64c3814
==== TABLE OF CONTENTS ====
The-trace-cone-classifies-Razak-Jacelon-stabilizations//figures/proof-roadmap.tex
The-trace-cone-classifies-Razak-Jacelon-stabilizations//paper.tex
The-trace-cone-classifies-Razak-Jacelon-stabilizations//preamble.tex
The-trace-cone-classifies-Razak-Jacelon-stabilizations//sections/01-introduction.tex
The-trace-cone-classifies-Razak-Jacelon-stabilizations//sections/02-structure.tex
The-trace-cone-classifies-Razak-Jacelon-stabilizations//sections/03-sequence-models.tex
The-trace-cone-classifies-Razak-Jacelon-stabilizations//sections/04-small-stabilizers.tex
The-trace-cone-classifies-Razak-Jacelon-stabilizations//sections/05-uniqueness.tex
The-trace-cone-classifies-Razak-Jacelon-stabilizations//sections/06-tracial-cone.tex
The-trace-cone-classifies-Razak-Jacelon-stabilizations//sections/07-cone-correction.tex
The-trace-cone-classifies-Razak-Jacelon-stabilizations//sections/08-point-intertwining.tex
The-trace-cone-classifies-Razak-Jacelon-stabilizations//references.bib
===========================
==== BEGIN The-trace-cone-classifies-Razak-Jacelon-stabilizations//figures/proof-roadmap.tex ====
\begin{figure}[tbp]
\centering
\begin{tikzpicture}[
stage/.style={draw=MidnightBlue!65,fill=MidnightBlue!3,
rounded corners=1pt,align=center,font=\small,
text width=54mm,minimum height=11mm,inner sep=3pt},
arrow/.style={-{Latex[length=2mm]},draw=MidnightBlue!80,thick}
]
\node[stage,text width=67mm] (data) at (0,0)
{Full trace cone\\Realification and ideal transport (Section~\ref{sec:structure})};
\node[stage,text width=67mm] (errors) at (0,-1.6)
{Fixed-cut comparisons and local error ideals\\(Section~\ref{sec:sequence-models})};
\node[stage] (small) at (-3.5,-3.3)
{Small stabilizers, with or without\\a given model (Section~\ref{sm:section})};
\node[stage] (rough) at (3.5,-3.3)
{Tracial approximation on the cone\\(Section~\ref{tc:section})};
\node[stage] (unique) at (-3.5,-5.0)
{Norm uniqueness for given models\\(Section~\ref{sec:uniqueness})};
\node[stage] (cone) at (3.5,-5.0)
{Fine-label correction and\\retained lower support (Section~\ref{cc:section})};
\node[stage,text width=67mm] (point) at (0,-6.8)
{Covariant line model and point model\\(Section~\ref{pt:section})};
\node[stage,text width=67mm] (actual) at (0,-8.4)
{Actual homomorphisms and intertwining\\(Section~\ref{pt:section})};
\draw[arrow] (data) -- (errors);
\draw[arrow] (errors.south west) -- (small.north);
\draw[arrow] (errors.south east) -- (rough.north);
\draw[arrow] (small) -- (unique);
\draw[arrow] (rough) -- (cone);
\draw[arrow] (small.east) -- (0,-3.3) -- (0,-5.0) -- (cone.west);
\draw[arrow] (unique.south) -- (point.north west);
\draw[arrow] (cone.south) -- (point.north east);
\draw[arrow] (point) -- (actual);
\draw[arrow,dashed] (unique.west) -- (-7.0,-5.0) -- (-7.0,-8.4) -- (actual.west);
\end{tikzpicture}
\caption{Principal model-construction dependencies. The existence and
uniqueness arguments meet in the passage from sequence models to actual
maps. Shared auxiliary inputs are not drawn separately: the trace-cap
and barycenter lemmas in Section~\ref{sec:uniqueness} are used in
Section~\ref{tc:section}; the absorption and hereditary-support lemmas
of Section~\ref{sec:uniqueness} are used in Section~\ref{cc:section}.
Norm uniqueness is a statement about
models when they are given; its proof does not assume the cone-existence
theorem. The dashed arrow records its further use in actual-map
extraction and the final two-sided intertwining.}
\label{fig:proof-roadmap}
\end{figure}
==== END The-trace-cone-classifies-Razak-Jacelon-stabilizations//figures/proof-roadmap.tex ====
==== BEGIN The-trace-cone-classifies-Razak-Jacelon-stabilizations//paper.tex ====
\documentclass[11pt]{article}
\input{preamble}
\title{The trace cone classifies Razak--Jacelon stabilizations}
\author{OpenAI}
\date{September 25, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove that the canonical topological cone of all extended
lower-semicontinuous tracial weights determines a separable nuclear
\(C^*\)-algebra after tensoring with the Razak--Jacelon algebra and the
compact operators, answering Robert's trace-cone classification question
positively. The isomorphism realizes the prescribed cone map, with
arbitrary ideal structure and without a density assumption on the finite
domains of the weights.
\end{abstract}
\tableofcontents
\input{sections/01-introduction}
\input{sections/02-structure}
\input{sections/03-sequence-models}
\input{sections/04-small-stabilizers}
\input{sections/05-uniqueness}
\input{sections/06-tracial-cone}
\input{sections/07-cone-correction}
\input{sections/08-point-intertwining}
\bibliographystyle{alphaurl}
\begingroup
\small
\raggedright
\bibliography{references}
\endgroup
\end{document}
==== END The-trace-cone-classifies-Razak-Jacelon-stabilizations//paper.tex ====
==== BEGIN The-trace-cone-classifies-Razak-Jacelon-stabilizations//preamble.tex ====
\input{glyphtounicode}
\input{glyphtounicode-cmex}
\pdfgentounicode=1
\ifdefined\pdfinfoomitdate\pdfinfoomitdate=1\fi
\ifdefined\pdftrailerid\pdftrailerid{}\fi
\ifdefined\pdfsuppressptexinfo\pdfsuppressptexinfo=15\fi
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[a4paper,margin=29mm]{geometry}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{aliascnt}
\usepackage{microtype}
\usepackage{needspace}
\usepackage{enumitem,booktabs,array}
\usepackage[dvipsnames]{xcolor}
\usepackage{xspace}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,positioning}
\usepackage{hyperref}
\hypersetup{
colorlinks=true,
linkcolor=MidnightBlue,
citecolor=MidnightBlue,
urlcolor=MidnightBlue,
pdftitle={The trace cone classifies Razak--Jacelon stabilizations},
pdfauthor={OpenAI}
}
\usepackage[capitalize,noabbrev,nameinlink]{cleveref}
\newcommand{\W}{\mathcal W}
\newcommand{\K}{\mathcal K}
\newcommand{\UHF}{\mathcal Q}
\newcommand{\C}{\mathbb C}
\newcommand{\R}{\mathbb R}
\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
\newcommand{\eps}{\varepsilon}
\newcommand{\cpc}{c.p.c.\xspace}
\DeclareMathOperator{\Cu}{Cu}
\DeclareMathOperator{\Prim}{Prim}
\DeclareMathOperator{\Her}{Her}
\DeclareMathOperator{\Ideal}{Ideal}
\DeclareMathOperator{\Fin}{Fin}
\DeclareMathOperator{\Ker}{Ker}
\DeclareMathOperator{\Ad}{Ad}
\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\KK}{KK}
\newtheorem{theorem}{Theorem}[section]
\newaliascnt{lemma}{theorem}
\newtheorem{lemma}[lemma]{Lemma}
\aliascntresetthe{lemma}
\newaliascnt{proposition}{theorem}
\newtheorem{proposition}[proposition]{Proposition}
\aliascntresetthe{proposition}
\newaliascnt{corollary}{theorem}
\newtheorem{corollary}[corollary]{Corollary}
\aliascntresetthe{corollary}
\theoremstyle{definition}
\newaliascnt{definition}{theorem}
\newtheorem{definition}[definition]{Definition}
\aliascntresetthe{definition}
\newaliascnt{convention}{theorem}
\newtheorem{convention}[convention]{Convention}
\aliascntresetthe{convention}
\theoremstyle{remark}
\newaliascnt{remark}{theorem}
\newtheorem{remark}[remark]{Remark}
\aliascntresetthe{remark}
\newaliascnt{example}{theorem}
\newtheorem{example}[example]{Example}
\aliascntresetthe{example}
\crefname{theorem}{Theorem}{Theorems}
\crefname{lemma}{Lemma}{Lemmas}
\crefname{proposition}{Proposition}{Propositions}
\crefname{corollary}{Corollary}{Corollaries}
\crefname{definition}{Definition}{Definitions}
\crefname{convention}{Convention}{Conventions}
\crefname{remark}{Remark}{Remarks}
\crefname{example}{Example}{Examples}
\crefname{section}{Section}{Sections}
\crefname{subsection}{Section}{Sections}
\crefname{equation}{Equation}{Equations}
\numberwithin{equation}{section}
\setcounter{tocdepth}{2}
\setcounter{secnumdepth}{2}
\setlist[enumerate]{leftmargin=*,itemsep=3pt,topsep=5pt}
\setlist[itemize]{leftmargin=*,itemsep=3pt,topsep=5pt}
\setlength{\parindent}{1.2em}
\setlength{\parskip}{0.12em}
\setlength{\emergencystretch}{1.5em}
\allowdisplaybreaks[2]
==== END The-trace-cone-classifies-Razak-Jacelon-stabilizations//preamble.tex ====
==== BEGIN The-trace-cone-classifies-Razak-Jacelon-stabilizations//sections/01-introduction.tex ====
\section{Introduction}\label{sec:introduction}
The Razak--Jacelon algebra \(\W\) is a simple, monotracial,
stably projectionless nuclear \(C^*\)-algebra
\cite{Jacelon}. Tensoring by \(\W\) removes the
\(K\)-theoretic part of the classification problem and makes the order
on the Cuntz semigroup visible through traces. The question is whether
the resulting trace data determine the algebra even when its ideal
structure is unrestricted. We prove that they do.
\subsection{The invariant and the main theorem}
For a \(C^*\)-algebra \(C\), let \(T(C)\) consist of all maps
\(\tau:C_+\to[0,\infty]\) satisfying
\[
\tau(0)=0,\qquad
\tau(a+b)=\tau(a)+\tau(b),\qquad
\tau(ra)=r\tau(a)\quad(r>0),
\]
together with \(\tau(x^*x)=\tau(xx^*)\) and lower semicontinuity for
norm-convergent nets in \(C_+\). Arithmetic is extended nonnegative
arithmetic. We impose no density condition on the finite domain.
In particular, every closed two-sided ideal \(I\) contributes the weight
\begin{equation}\label{intro:ideal-weight}
\tau_I(a)=
\begin{cases}
0,&a\in I_+,\\
\infty,&a\in C_+\setminus I_+.
\end{cases}
\end{equation}
The canonical topology is specified by
\begin{equation}\label{intro:trace-topology}
\tau_i\longrightarrow\tau
\quad\Longleftrightarrow\quad
\limsup_i\tau_i((a-\eps)_+)
\leq\tau(a)\leq\liminf_i\tau_i(a)
\quad(a\in C_+,\ \eps>0).
\end{equation}
An isomorphism of these topological cones is a homeomorphism preserving
addition, the zero weight, and multiplication by every positive real
scalar.
\Needspace{3\baselineskip}
\begin{theorem}\label{thm:main}
Let \(A\) and \(B\) be separable nuclear complex \(C^*\)-algebras.
If \(T(A)\) and \(T(B)\) are isomorphic as topological cones, then
\[
A\otimes\W\otimes\K\ \cong\ B\otimes\W\otimes\K,
\]
where \(\K=\K(\ell^2(\N))\) and all tensor products are spatial.
\end{theorem}
The conclusion is an actual \(C^*\)-algebra isomorphism.
The algebras may be nonunital, may have both finite and infinite
subquotients, and may have an arbitrary primitive ideal space.
Their only regularity assumptions are separability and nuclearity.
Moreover, the isomorphism can realize the prescribed cone map: after
transporting the given map to
\(F:T(B\otimes\W\otimes\K)\to T(A\otimes\W\otimes\K)\),
we obtain an isomorphism \(\Phi\) with
\(\tau\circ\Phi=F(\tau)\) for every extended trace \(\tau\).
This refinement is proved at the end of \Cref{pt:intertwining}.
The presence of \eqref{intro:ideal-weight} makes the invariant sensitive
to this ideal structure. For example,
\[
I\subseteq J
\quad\Longleftrightarrow\quad
\tau_I+\tau_J=\tau_I.
\]
The finite and zero ideals of a general weight are also encoded by the
two limiting operations obtained by letting its positive scalar
multiple tend to zero and to infinity. These features will let us
localize each approximation on the ideal where its controlling trace
is finite.
\subsection{History and significance}
Razak's classification of simple inductive limits of stably
projectionless building blocks made the cone of traces, together with
its scale, an effective classification invariant
\cite[Theorems~1.1 and~2.1]{Razak}. Jacelon's construction singled out
the simple monotracial algebra $\W$ in this setting \cite{Jacelon}.
The existence, uniqueness, and approximate-intertwining organization
of those classification arguments also provides the broad method used
here, with new local estimates needed for arbitrary ideals.
The trace-cone question is due to Leonel Robert. It was recorded in
Santiago's 2012 conference abstract \cite{SantiagoWStable} and appears
as Problem~LXVIII in the survey of Schafhauser, Tikuisis, and White
\cite{STW}. Their formulation retains every extended trace, including
the zero/infinity ideal weights. Their May 2026 discussion describes
the known simple and traceless cases and the difficulty already
presented by one proper nonzero ideal. \Cref{thm:main} gives a positive
answer to that problem for the full cone and the unrestricted class of
separable nuclear algebras in its statement.
Elliott, Robert, and Santiago developed the compact topological cone
of extended lower-semicontinuous traces and its relation to
functionals on the Cuntz semigroup \cite{ERS}.
Robert subsequently identified tensoring with $\W$ with
realification of that semigroup \cite[Theorem~5.1.2]{Robert}.
Thus the cone in \eqref{intro:trace-topology} determines the Cuntz
semigroup of the stabilized algebra, including its ideals and
positive real scalar multiplication. The remaining task is to realize
this invariant by algebra maps and prove their uniqueness.
In the tracial simple case, the work of Elliott and Niu \cite{ElliottNiu}
and the published classification of Elliott, Gong, Lin, and Niu
\cite[Theorem~7.5]{EGLN} establish classification by scaled traces
under the stated $KK$-contractibility and finite-nuclear-dimension
hypotheses. Castillejos and Evington's nuclear-dimension theorem for
simple separable nuclear $\mathcal Z$-stable algebras supplies the
regularity bridge for the simple stabilized case
\cite[Theorem~A]{CastillejosEvington}. Nawata gave another proof that
$A\otimes\W\cong\W$ when $A$ is simple, separable, nuclear, has a
unique tracial state, and has no unbounded densely defined traces
\cite[Corollary~6.2(i)]{Nawata}. These simple-algebra hypotheses are
not imposed on $A$ and $B$ in \Cref{thm:main}.
As a separate regularity consequence, every stabilization
$A\otimes\W\otimes\K$ in \Cref{thm:main} has
nuclear dimension at most one, by \cite[Theorem~1.3]{OpenAINuclearDimension}
together with $\W\otimes\mathcal Z\cong\W$ from
\cite[Corollary~6.3]{Jacelon}.
The classification argument below does not use this nuclear-dimension
bound.
The traceless case has a different structure: when every extended
trace takes only zero and infinity, the trace cone reduces to ideal
information. R\o rdam's absorption criterion
\cite[Theorem~5.2]{RordamZ} and Kirchberg's ideal-lattice classification,
with a complete proof in \cite[Theorem~6.13]{GabeO2}, explain the known
traceless branch. The present argument must also retain finite trace
data and distinguish the regions where a weight is finite from those
where it is infinite. That distinction is necessary when finite and
infinite subquotients occur in the same algebra.
Restricted nonsimple precedents include Lin and Ng's classification
of certain essential extensions by $\W$, with a universal coefficient
theorem assumption, a simple quotient, and an invariant containing
$K$-theory and trace data \cite[Theorem~9.6]{LinNg}.
The general cone-only question allows arbitrary ideals.
The main analytic tools have separate roles. Ciuperca, Giordano, Ng,
and Niu give unital von Neumann algebra uniqueness from support-rank data
\cite{CGNN}; we combine this with trace caps and rational averaging to
obtain local rank estimates. The stable-uniqueness methods of
Dadarlat and Eilers \cite{DadarlatEilers} and the ideal-related
absorption and multiplier results of Gabe
\cite[Sections~11--13]{Gabe} remove the resulting errors in norm.
Those intermediate results apply here without pure infiniteness of
the auxiliary targets. In the existence argument, Connes's
hyperfiniteness theorem \cite{Connes}, nuclear completely positive
approximation \cite{ChoiEffros}, and measured-field methods
\cite{VaesWouters} provide the classical inputs. The fixed controllers,
lower ideal-support estimates, and fine-label correction that connect
these tools are proved below.
The strategy of lifting tracial data and then correcting a map through
an extension has a methodological precedent in Schafhauser's proof of
the Tikuisis--White--Winter theorem \cite{Schafhauser}. His setting
uses a faithful amenable trace and the universal coefficient theorem
(UCT); the present local error
ideals and support estimates address a different invariant, with no
UCT assumption. The use of quotient cone approximations and carriers
in prescribed ideals also has a precedent in
Bosa--Gabe--Sims--White \cite[Section~3]{BGSW}. Here the small UHF
fractions must satisfy both upper trace bounds and lower comparisons
into fixed positive cuts. The suspension-and-corner passage follows
the method in \cite[Remark~6.4]{GabeO2} and
\cite[proof of Theorem~14.1]{Gabe}; its trace normalization is proved
below for all extended weights.
\subsection{The construction}
Fix
\[
P=A\otimes\W\otimes\K,\qquad
Q=B\otimes\W\otimes\K.
\]
For a $C^*$-algebra $C$, the Cuntz semigroup $\Cu(C)$ consists of
Cuntz classes $[a]$ of positive elements in $C\otimes\K$, with addition
by orthogonal sum. The relation $a\precsim b$ means that
$v_n^*bv_n\to a$ in norm for some sequence $(v_n)$; it determines the
order on these classes. Two elements define the same class when each
is Cuntz subequivalent to the other. For a positive contraction $a$, its rank at an
extended trace is $d_\tau(a)=\lim_k\tau(a^{1/k})$, using unnormalized
matrix traces. This depends only on $[a]$.
Two obstacles distinguish this setting from one with a single
normalized trace. A positive element generating a stable algebra may
have infinite rank at every nonzero densely finite trace, so making
its rank a small scalar multiple gives no useful finite error bound.
Also, a weight can be finite only on a proper ideal. We must control
errors separately on these ideals and preserve enough lower support
to retain the infinite values outside them.
The given cone isomorphism yields compatible isomorphisms
\[
F:T(Q)\longrightarrow T(P),\qquad
G:\Cu(P)\longrightarrow\Cu(Q),
\qquad
d_\tau(G[a])=d_{F\tau}(a).
\]
We identify the primitive ideal spaces through this correspondence
and denote the common space by \(X\). For an open $U\subseteq X$,
write $P(U)$ and $Q(U)$ for the corresponding closed ideals.
We first construct maps into a sequence algebra, where errors may tend
to zero along the coordinates. Norm uniqueness will later allow their
coordinate maps to be conjugated into convergent sequences in $Q$.
Our intermediate maps take values in
\[
D=Q_\infty=\ell^\infty(Q)/c_0(Q).
\]
We regard $Q$ as the constant sequences in $D$. If $\omega$ is a free
ultrafilter on $\N$ and $\tau_n\in T(Q)$, their regularized limit on
bounded positive sequences is
\[
\rho([(x_n)])=\sup_{\eps>0}\lim_{n\to\omega}
\tau_n((x_n-\eps)_+),\qquad x_n\geq0.
\]
Section~\ref{sec:sequence-models} proves that this defines an extended
trace. We use $Y$ equal to a point, a half-open interval, or $\R$,
with a full-support Radon measure $m$, finite on the interval.
For $\nu\in T(P)$, write $\Fin(\nu)$ for the closed ideal generated by the positive
elements on which $\nu$ is finite. The product trace $m\otimes\nu$
is the integrated densely finite trace on $C_0(Y)\otimes\Fin(\nu)$,
extended to be infinity on positive elements outside this ideal.
On positive elementary tensors it has value $(\int f\,dm)\nu(a)$,
with $0\cdot\infty=0$.
A \emph{model} is a homomorphism
\(p:C_0(Y)\otimes P\to D\) satisfying
\begin{equation}\label{intro:model}
\rho\circ p=m\otimes F(\rho|_Q)
\end{equation}
for every such regularized limit \(\rho\). The quantifier includes
nondensely finite weights. An ideal of $C_0(Y)\otimes P$ has a fine
support, an open subset of $Y\times X$; its projection to $X$ is the
coarse support used to index local estimates.
The main difficulty is to preserve both the finite and the infinite
parts of \eqref{intro:model}. Fix a regularized limit $\rho$, put
$\sigma=\rho|_Q$, and write $Q(V)=\Fin(\sigma)$. On
$C_0(Y)\otimes P(V)$, controlled trace moments will identify the
pullback of $\rho$. This alone does not prevent the pullback from
becoming finite on a larger ideal. A separate lower ideal comparison
rules that out. Once the finite ideal is exactly
$C_0(Y)\otimes P(V)$, both sides of \eqref{intro:model} are infinite
on every positive element outside it. This two-part recovery is the
reason for the following constructions.
\paragraph{Local error ideals and stabilizers.}
For each open \(U\subseteq X\), we define an ideal \(J(U)\subseteq D\)
using uniform bounds on the ranks of positive cuts.
The error at each cut is a scalar tending to zero times the rank of
a fixed positive controller $c$ satisfying
$c\precsim(b-\delta)_+$ for some $b\in Q(U)_+$ and $\delta>0$.
Such errors are invisible to every regularized weight that is densely
finite on \(Q(U)\).
We construct small homomorphisms whose values belong to these ideals
and dominate any prescribed countable family of smaller errors.
Compact containment of open ideals supplies a common controller
when different approximation stages are combined.
The two versions in Section~\ref{sm:section} have different inputs:
one takes a small fraction of an existing model; the other builds a
small cone homomorphism before any model is available.
\paragraph{Tracial approximation with lower ideal support.}
Convex separation and von Neumann algebra approximation construct
completely positive cone maps with the required first and second
trace moments. The construction simultaneously imposes norm tests
whose positive averages force lower rank comparisons.
The comparison constant is independent of the averaging
multiplicities. This lower signal determines the part of the cone
on which a weight must remain infinite.
Factor representations and their measurable assembly are used only
through cut tests controlled by fixed finite-rank constants.
\paragraph{Correction with all ideal labels retained.}
A small stabilizer places the multiplicative errors in a
sigma-unital hereditary target. Module Stinespring dilation and
absorption then correct the cone map.
The support estimate is imposed on the entire projected ideal of
each input. It therefore survives scaling of the cone variable and
makes every dilation weakly equivariant for the finer primitive-space
labels. Compact corrections can be compressed to finitely many terms
of the original diagonal repeat, with an explicit tail estimate.
A second small homomorphism then dominates the correction itself.
Adding it back retains the lower ideal comparison, so the finite and
infinite parts combine to give the full model equation in
Section~\ref{cc:section}.
\paragraph{Uniqueness and passage to actual maps.}
For two models with the same data, tracial alignment first makes their
difference locally small. A stabilizer drawn from a spare model then
allows ideal-related stable uniqueness to remove that difference in
norm. The spare is supplied by splitting each model into two UHF
halves and replacing the halves successively. This argument, in
Section~\ref{sec:uniqueness}, assumes that the two models are given;
it does not depend on the later existence theorem.
Once corrected cone models exist, an
exhaustion of the line gives models with Lebesgue data.
Translation covariance and a full crossed-product projection give
a model for \(P\) itself. A finite positive partition identity computes
the trace of this corner, including infinite values.
Uniqueness for every pair of subsequences gives eventual conjugacy
between any two sufficiently late coordinate maps. We use these
conjugacies to make a subsequence Cauchy in norm on each input; its
limit is an actual homomorphism $P\to Q$. Repeating the construction
in the other direction gives a map $Q\to P$. Their compositions have the
identity data, so approximate intertwining proves \Cref{thm:main}.
Figure~\ref{fig:proof-roadmap} displays these dependencies. In
particular, the uniqueness theorem is conditional on a pair of models;
it is available when the existence construction later supplies them.
\input{figures/proof-roadmap}
\subsection{Organization and conventions}
\Cref{st:realification,st:ideals} collect the structural input.
The sequence comparisons and local diagonalization are proved in
\Cref{seq:sandwich,seq:diagonal}. The stabilizer construction is
\Cref{sm:stabilizers}, and norm uniqueness is
\Cref{un:uniqueness}. The two existence stages are
\Cref{tc:rough-model,cc:cone-existence}; the final passage to an
isomorphism is \Cref{pt:intertwining}.
All algebras and Hilbert spaces are complex. A completely positive
contractive map is abbreviated c.p.c. Traces on finite matrix
amplifications use the unnormalized matrix trace.
The universal UHF algebra is denoted by \(\UHF\), to distinguish it
from the target algebra \(Q\). Rational diagonal averaging uses the
normalized UHF trace only in the averaging direction.
The notation \(V\Subset U\) denotes compact containment in the
lattice of open subsets of \(X\), without a Hausdorff assumption.
==== END The-trace-cone-classifies-Razak-Jacelon-stabilizations//sections/01-introduction.tex ====
==== BEGIN The-trace-cone-classifies-Razak-Jacelon-stabilizations//sections/02-structure.tex ====
\section{The invariant, its ideals, and matrix conventions}
\label{sec:structure}
The trace cone supplies three inputs to the construction: pointwise
comparison of Cuntz classes in \(Q\), a common lattice of ideals for
\(P\) and \(Q\), and positive elements that control finite collections
of rank estimates inside those ideals. We establish these inputs
before fixing the matrix identifications used in the approximations.
All later comparisons are obtained from the coordinate algebra
\(Q\); passage to a sequence algebra will require bounded witnesses.
\subsection{Ranks and realification}
For \(a\in(C\otimes\K)_+\), extend \(\tau\in T(C)\) to matrices by
the unnormalized matrix trace and put
\begin{equation}\label{st:rank}
d_\tau(a)=\lim_{k\to\infty}\tau(a^{1/k})
\qquad(0\leq a\leq1).
\end{equation}
Rescaling an element to be contractive does not change its rank.
The resulting function depends only on its Cuntz class. We write
\(d_\tau(x)\) also for the value at \(x\in\Cu(C)\).
The basic cut and integration identities are
\begin{equation}\label{st:rank-cuts}
d_\tau(a)=\sup_{\eps>0}d_\tau((a-\eps)_+),
\qquad
\tau(a)=\int_0^{\|a\|}d_\tau((a-t)_+)\,dt.
\end{equation}
These identities use extended nonnegative arithmetic and apply to all
the weights in \eqref{intro:trace-topology}.
We use the following form of the trace and realification theorems.
The assertion about the functional cone is
\cite[Proposition~4.2, Remark~4.3, and Theorem~4.4]{ERS}; compactness
is \cite[Theorem~3.7]{ERS}. The realification assertions are
\cite[Proposition~3.1.1 and Theorems~3.2.1 and~5.1.2]{Robert}.
\begin{theorem}[Trace realification]\label{st:realification}
For every \(C^*\)-algebra \(C\), the trace cone with topology
\eqref{intro:trace-topology} is compact Hausdorff, and it is second
countable when \(C\) is separable. If \(C\) is exact, the assignment
\(\tau\mapsto d_\tau\) identifies this cone with the topological
cone of functionals on \(\Cu(C)\).
For every \(C^*\)-algebra \(C\),
\[
\Cu(C\otimes\W)\cong\Cu(C)_{\R}.
\]
Realification is determined functorially by the topological functional
cone. Its realization as functions on that cone has pointwise order,
pointwise addition, and multiplication by positive real scalars.
It has the same functional cone as \(\Cu(C)\).
Consequently the given trace-cone isomorphism induces isomorphisms
\begin{equation}\label{st:transport}
F:T(Q)\longrightarrow T(P),\qquad
G:\Cu(P)\longrightarrow\Cu(Q),\qquad
d_\tau(Gx)=d_{F\tau}(x).
\end{equation}
In particular, for \(x,y\in\Cu(Q)\),
\begin{equation}\label{st:pointwise-order}
x\leq y
\quad\Longleftrightarrow\quad
d_\tau(x)\leq d_\tau(y)\quad\hbox{for every }\tau\in T(Q).
\end{equation}
Multiplication by every positive real scalar is an order automorphism
preserving increasing suprema and compact containment.
\end{theorem}
Here the function realization is Robert's cone
\(L(\operatorname{F}(\Cu(C)))\), where
\(\operatorname{F}(\Cu(C))\) denotes the functional cone. For a
functional cone \(T\), the cone \(L(T)\) consists of pointwise suprema
\(f=\sup_n f_n\), where the \(f_n:T\to[0,\infty]\) are additive,
positively homogeneous, lower semicontinuous, and zero at zero.
The approximating sequence satisfies, for some \(0<\eps_n<1\),
\[
f_n\leq(1-\eps_n)f_{n+1},
\qquad f_n\text{ is continuous at every }\tau\text{ with }f_{n+1}(\tau)<\infty.
\]
Continuity here is on all of \(T\) at the indicated point, with the
usual topology on \([0,\infty]\), rather than continuity only of a
restriction to the finite locus.
For the exact algebras considered below the realization is \(L(T(C))\).
This specified approximation condition is essential; arbitrary
lower-semicontinuous homogeneous functions are not being identified
with Cuntz classes. Composition with a cone homeomorphism preserves
all the displayed conditions, which is the functoriality used here. This gives \(G\) in
\eqref{st:transport}. We fix the normalized trace of \(\W\) when
identifying the trace cones before and after tensoring; stabilization
uses the unnormalized trace of \(\K\).
The equality of extended lower-semicontinuous quasitraces and traces
for exact algebras is included in the cited trace theorem. Its ancestry
is Haagerup's theorem for unital exact algebras
\cite[Theorem~5.11]{Haagerup} and the Blanchard--Kirchberg reduction
recalled in \cite[Remark~4.3]{ERS}. We use the extended nonunital
interface stated there. Thus the functional cone used here is exactly
the invariant of the problem.
We also record the tensor-factor properties
\begin{equation}\label{st:W-properties}
\W\otimes\UHF\cong\W,
\qquad
\W\otimes\W\cong\W,
\qquad
\KK(\W,\W)=0.
\end{equation}
The first property follows from
\cite[Proposition~6.2]{Jacelon}; the remaining properties are recorded
in \cite[Section~2.1 and Corollary~6.2]{Nawata}.
They are properties of the tensor factor \(\W\), so their use places
no simplicity restriction on \(A,B,P\), or \(Q\).
The UHF absorption is used in the foldings below, and the
ideal-related use of the last equality is proved in
\Cref{un:KK-nullity}.
For completeness, the tensor-square and approximate-inner-flip
properties underlying realification have accessible proofs in these
same sources. Choose $\beta:\W\otimes\W\to\W$ by
\cite[Corollary~6.2(i)]{Nawata}. Conjugating the flip by $\beta$
gives a trace-preserving automorphism of $\W$, which is approximately
inner by \cite[Corollary~4.6]{Jacelon}. Pulling the implementing
unitaries back through the unitized $\beta$ gives approximate inner
flip of $\W\otimes\W$, as in \cite[Remark~5.2]{Jacelon}.
Thus the tensor-square isomorphism and the approximate inner flip
needed for realification follow from properties of the factor itself.
\subsection{Finite ideals, zero ideals, and open supports}
For \(\tau\in T(C)\), let \(\Fin(\tau)\) be the closed ideal
generated by the positive elements on which \(\tau\) is finite,
and let \(\Ker(\tau)\) be its zero ideal. The restriction of
\(\tau\) to \(\Fin(\tau)\) is densely finite, whereas
\(\tau(a)=\infty\) for \(a\in C_+\setminus\Fin(\tau)\).
A useful characterization is
\begin{equation}\label{st:finite-rank-cuts}
a\in\Fin(\tau)
\quad\Longleftrightarrow\quad
d_\tau((a-\eps)_+)<\infty\quad\hbox{for every }\eps>0.
\end{equation}
To verify the forward direction, approximate \(a\) in norm by a
finite-trace positive element \(b\) of \(\Fin(\tau)\).
If \(\|a-b\|<\eps/2\), cut comparison gives
\[
d_\tau((a-\eps)_+)
\leq d_\tau((b-\eps/2)_+)
\leq \frac{2}{\eps}\tau(b)<\infty.
\]
The finite-trace positives are dense in this ideal: finite sums of
terms \(x^*bx\), with \(\tau(b)<\infty\), still have finite trace
and generate its positive cone densely. Conversely, finite rank of
all positive cuts implies finite trace of those cuts, since
\(\tau((a-\eps)_+)\leq\|a\|d_\tau((a-\eps)_+)\).
Their norm limit is \(a\), proving \eqref{st:finite-rank-cuts}.
\begin{proposition}[Transport of ideals]\label{st:ideals}
The maps \(F,G\) identify the complete ideal lattices of \(P\) and
\(Q\), and these identifications agree. They identify the finite
and zero ideals of corresponding weights. The resulting
homeomorphism of primitive ideal spaces will be used to regard
\(\Prim(P)=\Prim(Q)=X\). For an open \(U\subseteq X\), write
\(P(U)\) and \(Q(U)\) for the corresponding ideals. Then
\[
G(\Cu(P(U)))=\Cu(Q(U)),
\]
and
\[
\Fin(\tau)=Q(U)\ \Longleftrightarrow\ \Fin(F\tau)=P(U),
\qquad
\Ker(\tau)=Q(U)\ \Longleftrightarrow\ \Ker(F\tau)=P(U).
\]
\end{proposition}
\begin{proof}
The additive idempotents of the trace cone are exactly the ideal
weights \(\tau_I\). Indeed \(\tau+\tau=\tau\) forces every
value to be zero or infinity, and the zero set is a closed ideal.
Moreover
\[
I\subseteq J\quad\Longleftrightarrow\quad
\tau_I+\tau_J=\tau_I.
\]
Thus \(F\) determines an order isomorphism of ideal lattices.
If \(F(\tau_I)=\tau_J\), then
\[
[a]\in\Cu(J)
\quad\Longleftrightarrow\quad
d_{F\tau_I}(a)=0
\quad\Longleftrightarrow\quad
d_{\tau_I}(G[a])=0
\quad\Longleftrightarrow\quad
G[a]\in\Cu(I).
\]
This proves compatibility with \(G\).
The topology recovers the two other ideals through
\begin{equation}\label{st:scalar-limits}
\lim_{r\downarrow0}r\tau=\tau_{\Fin(\tau)},
\qquad
\lim_{r\uparrow\infty}r\tau=\tau_{\Ker(\tau)}.
\end{equation}
These are \cite[Equation~(3.4)]{ERS}. They can also be checked
directly in \eqref{intro:trace-topology}: all positive cuts of an
element of \(\Fin(\tau)\) have finite trace by
\eqref{st:finite-rank-cuts}, whereas an element outside that ideal
has infinite trace; for the second limit use whether its trace is
zero or positive. Continuity and positive homogeneity of \(F\)
therefore preserve both limits.
An ideal-lattice isomorphism preserves prime ideals and their
hull--kernel topology. In a separable \(C^*\)-algebra prime ideals
are primitive \cite[Corollary~1]{DixmierPrime}, which gives the
asserted homeomorphism.
Finally, the zero algebra is distinguished by the cone: every nonzero
algebra has the nonzero weight \(\tau_{\{0\}}\). We may henceforth
assume that \(P,Q\ne0\).
\end{proof}
For a positive element \(c\in Q(U)\), we say that \(c\) is
\emph{compactly Cuntz supported in \(Q(U)\)} if
\begin{equation}\label{st:compact-controller}
[c]\leq[(b-\delta)_+]
\quad\hbox{for some }b\in Q(U)_+,\ \delta>0.
\end{equation}
Equivalently, \([c]\) is below an element compactly contained in a
Cuntz class from \(Q(U)\). This concerns the class of \(c\), not
compactness of a subset of a possibly non-Hausdorff space.
The next lemma permits one controller to be chosen before an
approximation sequence is constructed. Its finite multiplicity may
depend on the element being controlled. Later we will compensate for
that multiplicity by choosing a sufficiently late coordinate.
\begin{lemma}[Countable supports and common controllers]
\label{st:controllers}
The open-set lattice of \(X\) has a countable basis closed under
finite unions such that every open \(U\) is the union of basis opens
\(V\Subset U\). If \(V\Subset U\), there is a compactly Cuntz
supported \(b_{V,U}\in Q(U)_+\) with
\[
Q(V)\subseteq\Ideal_Q(b_{V,U}).
\]
Every compactly Cuntz supported \(c\in Q(V)_+\) then satisfies
\begin{equation}\label{st:controller-multiple}
[c]\leq N[b_{V,U}]
\quad\hbox{for some finite }N\in\N.
\end{equation}
The element \(b_{V,U}\) is independent of \(c\); the integer
\(N\) need not be.
\end{lemma}
\begin{proof}
Choose a norm-dense sequence \((a_j)\) of positive contractions in
\(Q\). The supports of \((a_j-r)_+\), for positive rational
\(r\), form a countable open basis, after taking finite unions.
For example, to find such a support inside \(U\) and containing
\(x\in U\), choose \(a\in Q(U)_+\) nonzero in the quotient at
\(x\), and approximate it sufficiently closely by some \(a_j\).
A rational cut larger than the approximation error belongs to
\(Q(U)\), and a cut smaller than the quotient norm remains nonzero
at \(x\).
The support of \((a_j-r)_+\) is compactly contained, in the ideal
lattice, in the support of \((a_j-r/2)_+\). Indeed, if a directed
union of ideals contains the latter element, norm approximation
puts the more deeply cut element in one member of that union.
Choosing two rational cut levels inside the preceding construction
gives a basis open compactly contained in the prescribed \(U\).
Let \(h\) be strictly positive in \(Q(U)\). The ideals generated
by \((h-1/k)_+\) increase with dense union \(Q(U)\).
Compact containment gives \(k\) such that
\(Q(V)\subseteq\Ideal_Q((h-1/k)_+)\); take this cut as
\(b_{V,U}\). If \(c\) satisfies \eqref{st:compact-controller}
inside \(Q(V)\), then \([c]\ll[z]\) for some \(z\in Q(V)_+\).
The Cuntz-semigroup ideal generated by \([b_{V,U}]\) consists of
the elements below \(\sup_N N[b_{V,U}]\). Thus
\([z]\leq\sup_N N[b_{V,U}]\), and compact containment proves
\eqref{st:controller-multiple}.
\end{proof}
For the domains used below,
\[
S_Y=C_0(Y)\otimes P,
\qquad S_Y(U)=C_0(Y)\otimes P(U).
\]
Their primitive spaces are \(Y\times X\). For \(s\in S_Y\),
let \(U_s\) be the projection to \(X\) of the open support of
\(\Ideal_{S_Y}(s)\). Projection is open, and
\begin{equation}\label{st:smallest-label}
s\in S_Y(U)\quad\Longleftrightarrow\quad U_s\subseteq U.
\end{equation}
Thus every element has a smallest coarse label. This is the
lower-semicontinuity condition on the domain ideal assignment used
in absorption. The fine labels on \(S_Y\), when used later, are
its actual open subsets of \(Y\times X\).
\subsection{Folding and rational averaging}
There are two matrix operations in the proof. Stabilization adds
internal matrix corners, whose ranks add with no normalization.
Convex averaging instead uses a separate UHF factor, whose normalized
trace supplies the convex coefficients. The following identifications
allow both operations to have range \(Q\) while retaining this
distinction and aligning any already chosen finite tests.
\begin{lemma}[Trace-preserving folding]\label{st:folding}
Finite matrix amplifications, including amplifications whose sizes
vary with the coordinate, may be folded into \(Q\) so as to preserve
all trace and rank calculations with unnormalized internal matrix
traces. The folding of a fixed first corner can be made arbitrarily
close to the identity on any prescribed finite subset of \(Q\),
by a multiplier-unitary adjustment.
If \(h_1,\ldots,h_l\) are \cpc maps with values in finite matrix
amplifications of \(Q\), and \(\lambda_i>0\) are rational with
\(\sum_i\lambda_i=1\), their normalized diagonal average is a
\cpc map \(h\) into \(Q\) satisfying, for \(s\geq0\),
\begin{align}
\tau(h(s))&=\sum_i\lambda_i\tau(h_i(s)),
&\tau(h(s)^2)&=\sum_i\lambda_i\tau(h_i(s)^2),
\label{st:average-traces}\\
d_\tau((h(s)-\eps)_+)
&=\sum_i\lambda_i d_\tau((h_i(s)-\eps)_+).
\label{st:average-ranks}
\end{align}
The traces on the right include the unnormalized internal matrices.
All equalities include zero and infinite values.
\end{lemma}
\begin{proof}
Use \eqref{st:W-properties} once to exhibit
\(Q=C\otimes\UHF\otimes\K\). For a matrix amplification choose
a Hilbert-space unitary
\(U_k:\C^k\otimes\ell^2\to\ell^2\) and fold by
\(\id_{C\otimes\UHF}\otimes\Ad(U_k)\).
This preserves the unnormalized matrix extension of every trace.
For a finite set \(E\subset Q\), choose a finite-rank projection
\(e\) in the last factor such that
\(\|a-eae\|\) is small for \(a\in E\).
A unitary of \(B(\ell^2)\) can send the folded first-corner copy
of \(e\ell^2\) back to \(e\ell^2\), agreeing with the specified
identification there. Conjugating the folding by this unitary fixes
the compressed finite tests. The two compression errors give the
desired approximation on \(E\).
Write \(\lambda_i=l_i/L\). In an additional UHF factor choose
orthogonal projections \(e_i\) with normalized traces
\(l_i/L\) and sum one. After folding the internal matrices, form
\[
h(s)=\sum_i h_i(s)\otimes e_i.
\]
Orthogonality makes its norm the maximum of the block norms, and
functional calculus is blockwise for functions vanishing at zero.
This proves complete positivity, contractivity, and
\eqref{st:average-traces}--\eqref{st:average-ranks} before the final
UHF identification.
Choose that identification by interleaving tensor factors of a fixed
infinite tensor-product realization of \(\UHF\). Finite tensor
permutation unitaries identify its first-factor embedding with the
identity to any prescribed accuracy on finite tests. For a finite
matrix direction the trace identity on matrix units gives
\(\widetilde\tau(a\otimes e)=\operatorname{tr}(e)\tau(a)\),
where \(\tau(a)=\widetilde\tau(a\otimes1)\) and
\(\operatorname{tr}\) is normalized. Increasing finite matrix
algebras and positive cuts give this factorization for the UHF
direction, including extended weights. Thus the identification
preserves the displayed formulas even when \(Q\) is nonunital.
Multiplier conjugation also preserves every extended trace, by the
trace identity applied to \(a^{1/2}u\).
The choices are made separately at each coordinate; finite sets,
matrix sizes, and accuracies may therefore vary with that coordinate.
\end{proof}
We always distinguish the internal matrix directions from the
normalized candidate-averaging direction. In particular, an internal
sector sum is not divided by its number of sectors. All subsequent
padding and foldings use \Cref{st:folding} with the existing constant
and finite test elements included among the elements to be aligned.
==== END The-trace-cone-classifies-Razak-Jacelon-stabilizations//sections/02-structure.tex ====
==== BEGIN The-trace-cone-classifies-Razak-Jacelon-stabilizations//sections/03-sequence-models.tex ====
\section{Sequence models and local error ideals}\label{sec:sequence-models}
Put \(D=Q_\infty=\ell^\infty(Q)/c_0(Q)\), and identify \(Q\)
with its constant sequences. We use square brackets for sequence
classes when the distinction from coordinates matters. The two
purposes of this section are to turn trace identities into uniform
cut comparisons, and to retain local error bounds when approximation
stages and matrix sizes vary.
\subsection{Regularized limits and their model data}
\begin{proposition}[Regularized limits]\label{seq:regularized}
Let \(\omega\) be a free ultrafilter on \(\N\), and let
\(\tau_n\in T(Q)\) be arbitrary. For a bounded positive sequence
\((x_n)\), the formula
\begin{equation}\label{seq:regularization-formula}
\rho([(x_n)])=
\sup_{\eps>0}\lim_{n\to\omega}\tau_n((x_n-\eps)_+)
\end{equation}
defines a lower-semicontinuous tracial weight on \(D\).
Its rank functional is
\begin{equation}\label{seq:regularized-rank}
d_\rho([(x_n)])=
\sup_{\eps>0}\lim_{n\to\omega}d_{\tau_n}((x_n-\eps)_+).
\end{equation}
If \(\sigma=\lim_{n\to\omega}\tau_n\) in the canonical topology
on \(T(Q)\), then \(\rho|_Q=\sigma\).
\end{proposition}
\begin{proof}
Pull \(\tau_n\) back to \(\ell^\infty(Q)\) by its \(n\)-th
coordinate homomorphism. The compact Hausdorff trace cone has a
unique limit along \(\omega\), say \(\widetilde\rho\).
For every positive \(x\) in the product, the defining topology gives
\[
\lim_\omega\tau_n((x_n-\eps)_+)
\leq\widetilde\rho(x),\qquad
\widetilde\rho((x-\eps)_+)
\leq\lim_\omega\tau_n((x_n-\eps)_+).
\]
Taking suprema over \(\eps\) proves
\eqref{seq:regularization-formula} on the product.
Every fixed cut of an element of \(c_0(Q)_+\) is eventually zero,
so \(\widetilde\rho\) vanishes on that ideal and descends to a
lower-semicontinuous trace \(\rho\) on the quotient. This also
proves independence of the representative. The same two inequalities
for constant sequences give \(\rho|_Q=\sigma\).
For completeness, rescale the representatives so that
\(0\leq x_n\leq1\), and let the right side of
\eqref{seq:regularized-rank} be \(L\).
For every \(k\) and \(t>0\),
\[
\tau_n((x_n^{1/k}-t)_+)
\leq d_{\tau_n}((x_n-t^k)_+).
\]
Regularization and then the supremum over \(k\) show
\(d_\rho(x)\leq L\).
Conversely, choose a continuous function \(g_\eps\) with values in
\([0,1]\), zero on \([0,\eps/2]\), and one on
\([\eps,1]\). For \(0<t<1\),
\[
\tau_n((g_\eps(x_n)-t)_+)
\geq(1-t)d_{\tau_n}((x_n-\eps)_+).
\]
The support of \(g_\eps(x)\) is below the support of \(x\).
Thus \(d_\rho(x)\geq\rho(g_\eps(x))\), and regularization followed
by \(t\downarrow0\) gives
\(d_\rho(x)\geq\lim_\omega d_{\tau_n}((x_n-\eps)_+)\).
Taking the supremum over \(\eps\) proves the reverse inequality.
\end{proof}
We call the weights in \Cref{seq:regularized} \emph{regularized
limits}. The coordinate weights need not be bounded, normalized, or
densely finite. Passing to a subsequence is included by taking an
ultrafilter concentrated on its indices.
Let \(Y\) be a point, a half-open interval, or \(\R\), with a
full-support Radon measure \(m\). The measure is finite on a cone
interval and locally finite on \(\R\). Write \(S=S_Y\).
For \(\nu\in T(P)\), the product \(m\otimes\nu\) is the
integrated densely finite trace on
\(C_0(Y)\otimes\Fin(\nu)\), extended to be infinity on positives
outside that ideal. In particular,
\begin{equation}\label{seq:product-rank}
d_{m\otimes\nu}(f\otimes a)
=m(\{f>0\})d_\nu(a)
\qquad(f\in C_0(Y)_+,\ a\in P_+).
\end{equation}
Here a zero factor makes the product zero, including the expression
\(0\cdot\infty\). Full support of \(m\) ensures that a nonzero
positive continuous function has support of positive measure.
\begin{definition}[Model]\label{seq:model}
A model for \((Y,m,F)\) is a homomorphism \(p:S_Y\to D\) such that
\begin{equation}\label{seq:model-equation}
\rho\circ p=m\otimes F(\sigma),\qquad \sigma=\rho|_Q,
\end{equation}
for every regularized limit \(\rho\).
\end{definition}
We fix a countable collection of positive elementary tensors for
testing this identity. In each basis ideal \(P(V)\), choose a
countable dense positive set, include its positive rational cuts,
and include the less deeply cut elements needed as enlarged tests.
In \(C_0(Y)\), choose countable dense positive sets on a compact
exhaustion and their rational cuts, with enlarged compact supports.
Take all tensor products of these choices, including the zero tensor
and all positive rational cuts of each chosen factor. A cut above a
factor's norm is simply zero, and its rank identity is automatic.
For the line or a varying interval, a compact test and its enlarged support are chosen inside
one common open interval before an exhaustion index is advanced.
\begin{lemma}[Elementary determination]\label{seq:elementary-tests}
A trace \(\theta\in T(S_Y)\) equals \(m\otimes\nu\) if its
ranks on the preceding countable test collection satisfy
\eqref{seq:product-rank}. In particular, checking those ranks for
every regularized limit suffices for \eqref{seq:model-equation}.
\end{lemma}
\begin{proof}
We first extend the rank identities to all elementary tensors, then
identify the ideal where the weights are densely finite, and finally
compare their bounded restrictions inside that ideal.
\emph{Extension of the tests.}
Suppose that positive contractions \(f_j,a_i\) approximate
\(f,a\) within \(\eta\). The perturbation cut inequalities and
their tensor products give
\[
(f-3\eta)_+\otimes(a-3\eta)_+
\precsim(f_j-2\eta)_+\otimes(a_i-2\eta)_+
\precsim f\otimes a.
\]
The middle tensors belong to the test collection when \(\eta\) is
rational; their enlarged versions give the same comparison with any
fixed positive cuts of \(f\) and \(a\) on the left.
Ranks preserve increasing suprema of such cuts. Applying these
comparisons to both \(\theta\) and \(m\otimes\nu\), and letting
\(\eta\downarrow0\), proves \eqref{seq:product-rank} for every
elementary tensor with compact function support. Exhaustion gives
the remaining elementary tensors.
\emph{Identification of the finite ideal.}
Set \(I=\Fin(\nu)\). The finite ideal of \(\theta\) is exactly
\(C_0(Y)\otimes I\). For one inclusion, a compactly supported
function and a compactly Cuntz supported positive element of \(I\)
give a tensor of finite \(\theta\)-rank. Their ideals generate
\(C_0(Y)\otimes I\).
For the other inclusion, let an open rectangle \(V\times U\) be
contained in the support of \(\Fin(\theta)\). Take nonzero
\(f\in C_c(V)_+\) and cut-supported \(a\in P(U)_+\).
Choose \(f_0\in C_c(V)_+\) equal to one on \(\supp(f)\), and
\(b\in P(U)_+\), \(\delta>0\), with
\(a\precsim(b-\delta)_+\). Then
\[
f\otimes a\precsim(f_0\otimes b-\delta/2)_+.
\]
The larger tensor belongs to \(\Fin(\theta)\), so \(f\otimes a\)
is compactly Cuntz supported there and has finite \(\theta\)-rank by
\eqref{st:finite-rank-cuts}. Formula \eqref{seq:product-rank} and
\(0<m(f>0)<\infty\) imply \(d_\nu(a)<\infty\).
Such elements generate \(P(U)\), so \(P(U)\subseteq I\).
Rectangles exhaust the open support of any ideal, proving the claim.
\emph{Equality on the finite ideal.}
It remains to identify the densely finite traces on this common
ideal; there is nothing to prove if \(I=0\).
Choose a strictly positive \(h\in I\), put \(e_n=(h-1/n)_+\),
and choose increasing positive compactly supported functions
\(f_n\) whose open supports exhaust \(Y\). The hereditary algebras
\[
H_n=\Her(f_n\otimes e_n)
=C_0(\{f_n>0\})\otimes\Her(e_n)
\]
increase with dense union in \(C_0(Y)\otimes I\).
Both traces have finite support rank on \(H_n\), because
\(m(f_n>0)d_\nu(e_n)<\infty\). Their restrictions are bounded
and extend normally to \(H_n^{**}\).
For a positive elementary tensor in \(H_n\), two-variable spectral
integration and the already established elementary rank identities
give
\begin{align*}
\theta(f\otimes a)
&=\int_0^{\|f\|}\int_0^{\|a\|}
d_\theta((f-t)_+\otimes(a-u)_+)\,du\,dt\\
&=\int_0^{\|f\|}\int_0^{\|a\|}
m(f>t)d_\nu((a-u)_+)\,du\,dt
=\Bigl(\int_Y f\,dm\Bigr)\nu(a).
\end{align*}
The first integrand is the value of the normal extension on the
product of the two spectral projections. Tonelli's theorem applies
to these nonnegative functions. Boundedness and algebraic tensor
density now give equality of the two traces on \(H_n\).
Choose a positive contractive approximate unit \((v_n)\) of the
common ideal with each \(v_n\) in the union of the \(H_j\).
For every \(x\geq0\) in that ideal, \(v_nxv_n\) belongs to one
of these hereditary algebras, and
\[
\theta(v_nxv_n)
=\theta(x^{1/2}v_n^2x^{1/2})\leq\theta(x),
\qquad v_nxv_n\longrightarrow x.
\]
Lower semicontinuity gives
\(\theta(x)=\lim_n\theta(v_nxv_n)\), and the same argument holds
for \(m\otimes\nu\). Equality on the localizations therefore
gives equality on the common finite ideal.
Both weights are infinity outside it. Thus they agree everywhere.
\end{proof}
\subsection{Uniform comparison and cut sandwiches}
The following elementary estimate is the source of the uniformity
needed when rows of models are combined.
\begin{lemma}[Bounded witnesses into a fixed cut]
\label{seq:uniform-witness}
Let \(a,b\) be positive elements in a matrix algebra over any
\(C^*\)-algebra, and suppose that \(a\precsim(b-\delta)_+\),
where \(\delta>0\). For every \(\eta>0\) there is a comparison
element \(w\) such that
\begin{equation}\label{seq:witness-bound}
\|w^*bw-a\|<\eta,
\qquad
\|w\|^2\leq\frac{\|a\|+\eta}{\delta}.
\end{equation}
The bound is independent of the matrix size.
\end{lemma}
\begin{proof}
Choose \(v\) with
\(\|v^*(b-\delta)_+v-a\|<\eta\), and put
\[
g_\delta(t)=
\begin{cases}
\sqrt{(t-\delta)_+/t},&t>0,\\
0,&t=0,
\end{cases}
\qquad w=g_\delta(b)v.
\]
Then \(w^*bw=v^*(b-\delta)_+v\), while
\(g_\delta(b)^2\leq\delta^{-1}(b-\delta)_+\).
This proves \eqref{seq:witness-bound}.
\end{proof}
\begin{lemma}[Strict comparison from regularized limits]
\label{seq:strict-comparison}
Let \(x,y\in D_+\). If there is \(0<\alpha<1\) such that
\begin{equation}\label{seq:strict-rank-hypothesis}
d_\rho(x)\leq\alpha d_\rho(y)
\quad\hbox{for every regularized limit }\rho,
\end{equation}
then \(x\precsim y\). The same assertion holds after finite
matrix amplification and after coordinatewise folding.
\end{lemma}
\begin{proof}
Choose bounded positive representatives \(x_n,y_n\).
If comparison fails, there is \(\eps>0\) with
\((x-2\eps)_+\not\precsim y\).
For every fixed \(\delta>0\), there must be arbitrarily late
coordinates with
\begin{equation}\label{seq:failed-coordinate-comparison}
(x_n-\eps)_+\not\precsim(y_n-\delta)_+.
\end{equation}
Otherwise \Cref{seq:uniform-witness}, applied for each approximation
accuracy, supplies bounded sequences of witnesses showing
\((x-\eps)_+\precsim y\), a contradiction.
Choose increasing indices \(n_k\) where
\eqref{seq:failed-coordinate-comparison} holds for \(\delta=1/k\).
Pointwise order in \(\Cu(Q)\), from
\eqref{st:pointwise-order}, gives a trace whose rank on the source
is strictly larger than its rank on the target. The smaller rank is
finite. If the larger rank is finite, divide the trace by that rank;
if it is infinite, divide by one plus the smaller rank. Thus we may
choose \(\tau_k\) with
\[
d_{\tau_k}((x_{n_k}-\eps)_+)\geq1,
\qquad
d_{\tau_k}((y_{n_k}-1/k)_+)\leq1.
\]
Take a free ultrafilter on these indices and the corresponding
regularized limit. Formula \eqref{seq:regularized-rank} gives
\(d_\rho(x)\geq1\). For each fixed \(t>0\), the cut
\((y_{n_k}-t)_+\) is eventually below
\((y_{n_k}-1/k)_+\), so the same formula gives
\(d_\rho(y)\leq1\). This contradicts
\eqref{seq:strict-rank-hypothesis}. All comparisons used to select
\(\tau_k\) took place in \(Q\). Matrix folding preserves those
comparisons and ranks by \Cref{st:folding}.
\end{proof}
\begin{proposition}[Sandwich comparisons]\label{seq:sandwich}
Let \(p:S_Y\to D\) be a model, let \(s=f\otimes a\geq0\), and
suppose that \(0<m_0:=m(f>0)<\infty\). For \(r>0\), choose
\(c_r\in Q_+\) representing \(rG[a]\). Then
\begin{equation}\label{seq:uncut-sandwich}
c_r\precsim p(s)\quad(0<r<m_0),
\qquad
p(s)\precsim c_r\quad(r>m_0).
\end{equation}
The following versions have fixed target cuts.
If \(\eta,\delta>0\), \(0<r<m(f>\eta)\), and \(c\) represents
\(rG[(a-\delta)_+]\), then, for every \(0<\kappa<\eta\delta\),
\begin{equation}\label{seq:lower-trimmed-sandwich}
c\precsim(p(f\otimes a)-\kappa)_+.
\end{equation}
If \(r'>m_0\) and \(\eps>0\), then a representative \(b\) of
\(r'G[a]\) has a positive cut \((b-\kappa)_+\) such that
\begin{equation}\label{seq:upper-trimmed-sandwich}
(p(f\otimes a)-\eps)_+\precsim(b-\kappa)_+.
\end{equation}
The choices in these two displays can be made independently of
the model \(p\). The complete collection of such comparisons on
the countable compact tests characterizes the model equation.
\end{proposition}
\begin{proof}
For a regularized limit \(\rho\), put \(\sigma=\rho|_Q\) and
\(t=d_{F\sigma}(a)\). The ranks of \(c_r\) and \(p(s)\) are
\(rt\) and \(m_0t\). Their ratio has the required strict scalar
slack in the appropriate direction, so
\Cref{seq:strict-comparison} proves \eqref{seq:uncut-sandwich}.
This reasoning includes \(t=0\) and \(t=\infty\); the separating
normalization in that lemma forces a finite nonzero value whenever
it is needed for a contradiction.
Apply the lower comparison to
\(s'=(f-\eta)_+\otimes(a-\delta)_+\).
Its support lies in the spectral region \(f\otimes a>\eta\delta\),
so elementary functional calculus gives
\(s'\precsim(f\otimes a-\kappa)_+\) whenever
\(0<\kappa<\eta\delta\). Applying \(p\) proves
\eqref{seq:lower-trimmed-sandwich}.
For the upper comparison, choose
\(m_0<r<r'\) and \(\delta>0\) with
\(\|f\|\delta<\eps/2\). The norm perturbation from \(f\otimes a\)
to \(f\otimes(a-\delta)_+\) gives
\[
(p(f\otimes a)-\eps)_+
\precsim p(f\otimes(a-\delta)_+)
\precsim c,
\qquad [c]=rG[(a-\delta)_+].
\]
Since positive scalar multiplication preserves compact containment,
\[
[c]\ll rG[a]\leq r'G[a]=[b].
\]
There is therefore \(\kappa>0\) with
\([c]\leq[(b-\kappa)_+]\). This choice uses only the indicated
constant classes, not \(p\).
Conversely, apply any regularized rank to the fixed-cut comparisons.
Let \(\delta,\eta\downarrow0\) and let the rational lower and
upper scalars approach \(m_0\). The resulting inequalities give
\eqref{seq:product-rank} for the pullback trace on every compact
test. Supremum of cuts handles zero or infinite ranks, and compact
exhaustion handles infinite-measure supports.
\Cref{seq:elementary-tests} now gives
\eqref{seq:model-equation}.
\end{proof}
\begin{corollary}[Constant ideals and all traces]\label{seq:all-traces}
For a model \(p\) and every \(s\in S\),
\begin{equation}\label{seq:constant-ideal}
\Ideal_D(p(s))=\Ideal_D(Q(U_s)).
\end{equation}
Given countably many models \(p_i:S\to D\), there is a separable
subalgebra \(D_0\subseteq D\) containing their ranges and the
constants \(Q\), such that, for every \(\gamma\in T(D_0)\),
\begin{equation}\label{seq:all-trace-model-equation}
\gamma\circ p_i=m\otimes F(\gamma|_Q)
\quad\hbox{for every }i.
\end{equation}
It may also contain any prescribed countable subset of \(D\).
\end{corollary}
\begin{proof}
For a nonzero elementary tensor of finite function support, the
upper sandwiches put its image in the ideal generated by the
corresponding constants. The lower trimmed sandwiches recover the
ideals of all cuts of \(a\), hence the whole constant ideal
\(Q(U_a)\).
For general \(s\), the ideal generated by \(s\) in \(S\) is
generated by positive elementary tensors whose rectangle supports
lie in its open support. Their projected supports exhaust \(U_s\).
Applying the elementary lower bounds gives
\(\Ideal_D(Q(U_s))\subseteq\Ideal_D(p(s))\).
Approximating \(s\) by finite sums of elementary tensors from
\(S(U_s)\) and applying the upper bounds gives the reverse inclusion.
For the second assertion, start with the separable algebra generated
by the stated ranges, constants, and extra elements. For every
countable test, every rational cut and scalar in
\Cref{seq:sandwich}, and every model, adjoin the matrix entries of
comparison witnesses at accuracies tending to zero. The resulting
algebra \(D_0\) is separable, and all these comparisons hold in
\(\Cu(D_0)\), not just in \(\Cu(D)\).
For \(\gamma\in T(D_0)\), write \(\sigma=\gamma|_Q\).
The rank of a constant representing \(rG[a]\) is
\(r d_{F\sigma}(a)\). Applying \(d_\gamma\) to the retained
sandwiches therefore gives the elementary rank formula for each
\(\gamma\circ p_i\). The proof of the converse in
\Cref{seq:sandwich} and \Cref{seq:elementary-tests} applies without
requiring \(\gamma\) to be a regularized limit. This proves
\eqref{seq:all-trace-model-equation}.
\end{proof}
\subsection{The ideals of locally small elements}
\begin{definition}[Local error ideal]\label{seq:smallideal}
For an open \(U\subseteq X\), an element \(x=[(x_n)]\in D\)
belongs to \(J(U)\) if, for every \(\eps>0\), there are a positive
element \(c\) compactly Cuntz supported in \(Q(U)\), a sequence
\(r_n>0\) tending to zero, and an index \(N\), such that
\begin{equation}\label{seq:smallideal-bound}
d_\tau((|x_n|-\eps)_+)
\leq r_n d_\tau(c)
\qquad(n\geq N,\ \tau\in T(Q)).
\end{equation}
The controller \(c\), the decay sequence, and \(N\) may depend on
\(x,U,\eps\). Once these have been chosen, the same bound holds
for all traces and all coordinates in the tail.
\end{definition}
The scalar \(r_n\) is strictly positive; consequently the right
side of \eqref{seq:smallideal-bound} is unambiguously infinity
when \(d_\tau(c)=\infty\). No estimate on the uncut element is
part of this definition.
For example, fix a nonzero positive contraction \(a\) compactly
Cuntz supported in \(Q(U)\). In an additional UHF factor choose
projections \(e_n\) of positive rational traces \(t_n\to0\).
After the trace-preserving identification of \Cref{st:folding},
the elements \(x_n=a\otimes e_n\) satisfy
\[
\|x_n\|=\|a\|,\qquad
d_\tau((x_n-\eps)_+)
=t_n d_\tau((a-\eps)_+)\leq t_n d_\tau(a).
\]
Thus \([(x_n)]\) belongs to \(J(U)\) and is nonzero. This is the
kind of smallness that allows a stabilizing map to retain its norm
while the coefficient in its local rank bound tends to zero.
\begin{proposition}[Properties of the local error ideals]
\label{seq:smallideal-properties}
The definition of \(J(U)\) is independent of representatives and
makes it a closed two-sided ideal of \(D\). The assignment is
order preserving, and
\begin{equation}\label{seq:smallideal-support}
J(U)\subseteq\Ideal_D(Q(U)).
\end{equation}
If \(\rho\) is a regularized limit and its restriction
\(\sigma=\rho|_Q\) is densely finite on \(Q(U)\), then
\begin{equation}\label{seq:smallideal-vanishing}
\rho|_{J(U)}=0.
\end{equation}
All assertions use the same conventions in finite matrix algebras.
\end{proposition}
\begin{proof}
We recall the cut estimates that will be used. For bounded operators
in a \(C^*\)-algebra and every tracial rank, the following weaker
forms of the spectral cut inequalities suffice:
\begin{align}
d_\tau((|x+y|-\eps)_+)
&\leq d_\tau((|x|-\eps/4)_+)
+d_\tau((|y|-\eps/4)_+),
\label{seq:sum-cut}\\
d_\tau((|bx|-\eps)_+),\ d_\tau((|xb|-\eps)_+)
&\leq d_\tau((|x|-\eps/(2M))_+)
\quad(\|b\|\leq M, M>0).
\label{seq:product-cut}
\end{align}
Also \(d_\tau((|x^*|-t)_+)=d_\tau((|x|-t)_+)\), and
\begin{equation}\label{seq:perturbation-cut}
\|x-y\|<\eps/4
\quad\Longrightarrow\quad
d_\tau((|x|-\eps)_+)
\leq d_\tau((|y|-\eps/2)_+).
\end{equation}
Here is an explicit cut reduction. Set
\[
x_t=x f_t(|x|),\qquad
f_t(s)=\begin{cases}(s-t)_+/s,&s>0,\\0,&s=0.\end{cases}
\]
Then \(\|x-x_t\|\leq t\) and
\(|x_t|=(|x|-t)_+\). Use the perturbation comparison
\(\|z-w\|<\eps\Rightarrow(|z|-\eps)_+\precsim|w|\).
For the sum take \(w=x_{\eps/4}+y_{\eps/4}\), and use
\(|w|\precsim|x_{\eps/4}|\oplus|y_{\eps/4}|\).
For a product take \(w=bx_{\eps/(2M)}\), whose absolute value is
Cuntz below \(|x_{\eps/(2M)}|\). For
\eqref{seq:perturbation-cut} take \(w=y_{\eps/2}\), so that
\(\|x-w\|<3\eps/4\). The identities for adjoints give the other
product order. These comparisons follow from the positive cut
perturbation lemma and \([v^*v]=[vv^*]\); applying the additive
rank functional proves the displays for extended weights as well.
If \(x,y\in J(U)\), choose their controllers at tolerance
\(\eps/4\). A folded finite block sum of the controllers is
compactly Cuntz supported in \(Q(U)\). Taking the maximum of the
two decay sequences proves the bound for \(x+y\) by
\eqref{seq:sum-cut}. Equation \eqref{seq:product-cut} proves
closure under multiplication on either side by an arbitrary bounded
sequence. The equality for adjoints proves self-adjointness.
Equation \eqref{seq:perturbation-cut} proves independence of the
representative. For norm closure, first fix \(\eps\), then choose
one element of the approximating sequence within \(\eps/8\) in
quotient norm. Its representatives are eventually within
\(\eps/4\), so its controller at tolerance \(\eps/2\) works for
the desired output cut. Monotonicity in \(U\) follows directly from
the definition.
To prove \eqref{seq:smallideal-support}, fix a cut of \(|x|\),
choose its controller \(c\precsim(b-\delta)_+\) in \(Q(U)\),
and wait until \(r_n\leq1\).
Pointwise order in \(\Cu(Q)\) gives
\((|x_n|-\eps)_+\precsim c\precsim(b-\delta)_+\).
The bounded witnesses of \Cref{seq:uniform-witness} give comparison
with the constant \(b\) in the quotient. Every positive cut of
\(|x|\) is consequently in \(\Ideal_D(Q(U))\), and so is \(x\).
For vanishing, fix \(\eps\) and the same type of controller.
For each coordinate weight,
\begin{equation}\label{seq:controller-trace-bound}
d_{\tau_n}(c)
\leq\frac4\delta\,
\tau_n((b-3\delta/4)_+).
\end{equation}
Indeed, on the spectral region \(b>\delta\), the function
\((b-3\delta/4)_+\) is at least \(\delta/4\).
Put \(h=(b-\delta/2)_+\). Dense finiteness on \(Q(U)\) and
\eqref{st:finite-rank-cuts} give \(\sigma(h)<\infty\), while
the trace-cone convergence inequalities give
\[
\lim_\omega\tau_n((b-3\delta/4)_+)
=\lim_\omega\tau_n((h-\delta/4)_+)
\leq\sigma(h).
\]
The ranks in \eqref{seq:controller-trace-bound} are therefore bounded
on an \(\omega\)-large set. Multiplying by \(r_n\to0\) in
\eqref{seq:smallideal-bound} and using
\eqref{seq:regularized-rank} yields \(d_\rho(|x|)=0\).
Hence \(\rho\) vanishes on \(J(U)_+\).
The proof uses only finite matrix sums and unnormalized ranks, so it
also proves the matrix assertions, with foldings as in
\Cref{st:folding}.
\end{proof}
\subsection{A diagonalization with fixed controllers}
The diagonalizations used below retain two different kinds of control.
A fixed positive controller bounds a local error's rank, while a
fixed positive target cut bounds the norm of a comparison witness.
Both must be chosen before advancing through the approximation
stages. We call each stage's sequence of coordinate approximations
a \emph{row}. Its finite matrix size may depend on the row and on
the coordinate; all matrices are folded by \Cref{st:folding}.
\begin{lemma}[Local diagonalization]\label{seq:diagonal}
Let \(\mathcal A\) be a countable dense
\(\mathbb Q+i\mathbb Q\) \(*\)-subalgebra of \(S=S_Y\),
containing the localized dense sets to be tested.
For each \(j\), let \(L_j:\mathcal A\to D\) be a map, and choose
bounded coordinate representatives \(L_{j,n}(a)\) for every
\(a\in\mathcal A\). Assume that, for each fixed input,
\[
\limsup_{j\to\infty}\|L_j(a)\|\leq\|a\|,
\]
and that the quotient errors in every fixed rational linearity,
involution, and multiplicativity relation tend to zero as
\(j\to\infty\). Only bounds on each fixed input are required.
Then one can choose \(j(n)\to\infty\) such that
\[
L(a)=[(L_{j(n),n}(a))]
\]
extends to a contractive homomorphism \(L:S\to D\).
The same diagonal choice can retain the following countable data.
\begin{enumerate}[label=\textup{(\roman*)}]
\item For each \(l\), fix basis opens \(V_l\Subset U_l\) and
row errors \(e_l^{(j)}\in J(V_l)\), for all sufficiently large
\(j\), with chosen representatives \(e_{l,n}^{(j)}\).
Assume \(\sup_j\|e_l^{(j)}\|<\infty\).
Then their diagonal \(e_l=[(e_{l,n}^{(j(n))})]\) belongs to
\(J(U_l)\).
\item For each prescribed comparison, let
\(a^{(j)},b^{(j)}\in D_+\) have uniformly bounded norms and
chosen bounded positive representatives. Suppose
\(a^{(j)}\precsim(b^{(j)}-\delta)_+\), where \(\delta>0\)
is independent of \(j\). If \(a,b\) are the respective
diagonals, then \(a\precsim(b-\delta')_+\) for any
preselected \(0<\delta'<\delta\).
\end{enumerate}
For local errors arising from maps on \(S\), it suffices to
schedule dense localized inputs in these pairs \(V\Subset U\).
The conclusion then extends to every open \(U\) and every input
in \(S(U)\) if the coordinate error maps are uniformly equicontinuous
on bounded sets, or if their diagonals are the values of a continuous
expression in the resulting quotient maps. This includes differences
from fixed equicontinuous coordinate maps and continuous multilinear
errors, with the other variables on countable dense lists.
In particular, a diagonal of models is again a model when their
data agree on the fixed compact tests and their enlarged versions.
This also applies to models on increasing intervals with unchanged
measure on these tests. Every subsequence of a model retains its
model data.
\end{lemma}
\begin{proof}
\emph{The common controller and the coordinate choice.}
Enumerate all algebraic relations and positive rational cut tests.
Reindex the rows so that row \(j\) includes the first \(j\)
requirements and their quotient norm errors tend to zero.
Include the bounds on each prescribed error and comparison element
among these requirements.
Set any as yet unspecified early-row errors to zero.
In the following construction let \(l\) enumerate pairs consisting
of a prescribed local error and a positive rational tolerance
\(\eps_l\); repeat that error's opens and representatives with
each tolerance.
Choose once and for all the controller
\(b_l=b_{V_l,U_l}\) from \Cref{st:controllers}.
In row \(j\), the \(l\)-th error has, after folding, a bound
\[
d_\tau((|e^{(j)}_{l,n}|-\eps_l)_+)
\leq r^{(j)}_{l,n}d_\tau(c^{(j)}_l),
\qquad r^{(j)}_{l,n}\longrightarrow0,
\]
uniformly for \(\tau\in T(Q)\) eventually in \(n\).
The controller \(c^{(j)}_l\) is compactly Cuntz supported in
\(Q(V_l)\). There is therefore a finite integer \(N_{j,l}\) with
\[
[c^{(j)}_l]\leq N_{j,l}[b_l].
\]
This integer absorbs internal matrix sizes, repeated blocks, and
any other finite multiplicity at that stage. No bound on it as
\(j\to\infty\) is required.
Fix a sequence \(\eta_j\downarrow0\). After the controllers and
the integers \(N_{j,l}\) have been chosen for \(l\leq j\),
choose strictly increasing coordinate thresholds \(n_j\geq j\)
so large that
\begin{equation}\label{seq:diagonal-delay}
N_{j,l}r^{(j)}_{l,n}\leq\eta_j
\qquad(n\geq n_j,\ l\leq j).
\end{equation}
At the same threshold impose the first \(j\) norm and algebraic
requirements, with an additional coordinate error at most
\(\eta_j\). This is possible because a quotient norm is the
limsup of the coordinate norms. Choose representatives for the
countable values; linearity of those representatives is unnecessary,
since the linearity relations themselves are among the tests.
For \(n_j\leq n<n_{j+1}\), put \(j(n)=j\) and use row \(j\).
The diagonal norm bounds make each prescribed value a bounded
sequence. For any fixed local
test \(l\), \eqref{seq:diagonal-delay} then gives, eventually,
\[
d_\tau((|e_{l,n}|-\eps_l)_+)
\leq\eta_{j(n)}d_\tau(b_l),
\qquad j(n)\longrightarrow\infty.
\]
The controller on the right is fixed, so this is precisely the
required \(J(U_l)\) bound. Enumerating all rational cut
tolerances proves membership in that ideal. The diagonal norm
requirements give \(\|L(a)\|\leq\|a\|\), and all algebraic
errors vanish. Thus the quotient values extend uniquely to a
contractive homomorphism on \(S\).
\emph{Extension of local errors.}
For an arbitrary open \(U\), the ideals \(S(V)\), with basis
opens \(V\Subset U'\subseteq U\), have dense union in \(S(U)\).
Indeed they generate \(S(U)\) as a closed sum, and finite sums
already lie in one member: replace finitely many pairs
\(V_i\Subset U'_i\) by their finite unions, using the basis
from \Cref{st:controllers}. Thus the union of the scheduled
localized dense sets is dense in \(S(U)\). Each retained error
on those sets lies in \(J(U')\subseteq J(U)\).
Closedness of \(J(U)\) and the assumed continuity extend this
containment to every localized input. For errors with several
variables, approximate each variable on its scheduled dense set,
using the global lists for unrestricted variables.
The continuity used here is either uniform equicontinuity of the
coordinate errors or continuity of their expression in the completed
quotient maps. Continuity of each row separately would not suffice.
\emph{Comparison witnesses with a fixed target cut.}
Suppose a row
comparison has uniformly bounded source \(a^{(j)}\) and uniformly
bounded \(b^{(j)}\), with target \((b^{(j)}-\delta)_+\) and
\(\delta>0\) independent of \(j\). Include both source and target
norms in the coordinate requirements, so their diagonals belong to
the sequence algebra.
For a chosen positive source cut and comparison accuracy, use
\Cref{seq:uniform-witness} with uncut target
\((b^{(j)}-\delta')_+\) and cut size \(\delta-\delta'\).
Its squared norm is bounded by the source norm plus the accuracy,
divided by \(\delta-\delta'\). Lift the witness to a
bounded coordinate sequence with arbitrarily small additional norm
and equation errors, and put its finite-test equation among the
requirements imposed at \(n_j\). This retained target slack gives
bounded witnesses on the diagonal. Enumerating source cuts and
accuracies proves the desired Cuntz comparisons there. Thus the
order of choices is: fixed tests and target cuts, row controllers
and witnesses, comparison multiplicities, and finally the coordinate
thresholds.
For rows of models, the lower bounds
\eqref{seq:lower-trimmed-sandwich} use target thresholds
\(\kappa<\eta\delta\) determined by fixed trimmed input factors.
The upper bounds \eqref{seq:upper-trimmed-sandwich} use cuts of fixed
constant representatives, determined by the scalar slack and the
fixed source tolerance. Both kinds of target threshold are
independent of the row. They can therefore be retained by the
preceding procedure. The converse part of \Cref{seq:sandwich}
identifies the diagonal model data. On increasing intervals the
same argument uses the common compact tests and their unchanged
measure; no condition is imposed on the moving endpoints.
Restriction to any subsequence retains all fixed-cut comparisons,
which proves the final assertion.
\end{proof}
\begin{remark}\label{seq:lifts}
When the resulting quotient map is a homomorphism with nuclear
separable domain, the completely positive lifting theorem provides
\cpc coordinate lifts \cite[Theorem~3.10]{ChoiEffrosLifting}. Those lifts represent the
same quotient values. Consequently all fixed-cut comparisons and
all memberships in \(J(U)\) are unchanged. This permits the
diagonal construction to be carried out first on countable algebraic
data, while later arguments use actual \cpc coordinate maps.
\end{remark}
==== END The-trace-cone-classifies-Razak-Jacelon-stabilizations//sections/03-sequence-models.tex ====
==== BEGIN The-trace-cone-classifies-Razak-Jacelon-stabilizations//sections/04-small-stabilizers.tex ====
\section{Small stabilizers}\label{sm:section}
We construct homomorphisms whose values belong to the local error
ideals, but are large enough to absorb a prescribed countable family
of errors. All matrix comparisons in this section are stable
comparisons. The identifications of
\Cref{st:folding} are understood; in particular, a rational UHF
fraction changes ranks, not operator norms.
\begin{theorem}[Small stabilizers]\label{sm:stabilizers}
Let \(S=C_0(Y)\otimes P\), and prescribe countably many pairs
\[
x_l\in S_+,
\qquad e_l\in J(U_{x_l}),
\qquad l\in\N.
\]
There is a homomorphism \(\theta:S\to D\) satisfying
\begin{equation}\label{sm:local-and-lower}
\theta(S(U))\subseteq J(U),
\qquad
e_l\in\Ideal_D(\theta(x_l))
\quad(U\subseteq X\text{ open},\ l\in\N),
\end{equation}
in either of the following situations.
\begin{enumerate}[label=\textup{(\roman*)}]
\item An existing model \(\chi:S\to D\) is given.
In this case \(\theta\) can be a slowly vanishing rational UHF
fraction of \(\chi\), and it also satisfies
\begin{equation}\label{sm:coarse-fullness}
\theta(S(U_x))\subseteq\Ideal_D(\theta(x))
\qquad(x\in S_+).
\end{equation}
For every fixed \(N\), \(N\) copies of its defining fraction
fit in the defining copy of \(\chi\) at all sufficiently late
coordinates.
\item \(Y=(0,1]\). No model is needed in this case, and
\eqref{sm:coarse-fullness} is not asserted.
\end{enumerate}
In both cases, for every \(l\) and \(\eps>0\) there is
\(\delta_{l,\eps}>0\) such that
\begin{equation}\label{sm:fixed-lower-cut}
[ (|e_l|-\eps)_+]
\leq[(\theta(x_l)-\delta_{l,\eps})_+]
\quad\text{in }\Cu(D).
\end{equation}
The construction permits the prescribed errors and the stabilizer
to occupy separate matrix corners.
\end{theorem}
The cut in \eqref{sm:fixed-lower-cut} will remain fixed while the
accuracy, matrix sizes, and finite lists change. We first describe
the countable tests and the comparison convention used to ensure
this.
\subsection{Fixed tests and comparison bounds}
Choose a countable compact-containment basis for the open ideals
as in \Cref{st:controllers}. For every pair \(V\Subset U\) in this
basis, choose a norm-dense sequence of positive contractions in
\(S(V)\), and include all positive rational cut tolerances.
Enumerate the resulting upper tests as
\begin{equation}\label{sm:upper-tests}
(a_i,\eps_i,V_i,U_i),\qquad
a_i\in S(V_i)_+,\quad V_i\Subset U_i,
\quad \eps_i>0.
\end{equation}
For each test fix a compactly Cuntz-supported positive
\(b_i\in Q(U_i)\) whose generated ideal contains \(Q(V_i)\).
These controllers are chosen once, before the approximation
stages. Showing
\begin{equation}\label{sm:upper-goal}
d_\tau((\theta_n(a_i)-\eps_i)_+)
\leq s_{i,n}d_\tau(b_i),
\qquad s_{i,n}\longrightarrow0,
\end{equation}
uniformly in \(\tau\), suffices for local smallness.
Indeed, \Cref{seq:smallideal-properties} first gives the conclusion
on the dense positive tests in \(S(V_i)\). Linearity and norm
closure then give it on \(S(V_i)\), and the ideals with
\(V_i\Subset U\) exhaust \(S(U)\).
Choose bounded representatives \(e_{l,n}\) for the errors.
Enumerate every error together with a sequence of positive cut
tolerances decreasing to zero. Write a lower request as
\(q=(l,\eta)\), and put
\begin{equation}\label{sm:error-controllers}
A_{q,n}=(|e_{l,n}|-\eta)_+,
\qquad
d_\tau(A_{q,n})\leq r_q(n)d_\tau(c_q),
\qquad r_q(n)>0,\quad r_q(n)\longrightarrow0.
\end{equation}
Here \(c_q\) is a fixed compactly Cuntz-supported positive
in \(Q(U_{x_l})\), and the inequality holds eventually for
every \(\tau\in T(Q)\). Enlarging finitely many values of
\(r_q\) will not matter. A request with zero controller has
eventually zero source cut and can be omitted.
We will use the fixed-cut comparison bound from
\Cref{seq:uniform-witness}. Its elementary form is worth recording.
If \(A\precsim(B-\delta)_+\), choose \(v\) with
\(v^*(B-\delta)_+v\) close to \(A\), and set
\[
w=g_\delta(B)v,
\qquad
g_\delta(t)=\sqrt{\frac{(t-\delta)_+}{t}},
\qquad g_\delta(0)=0.
\]
Then
\begin{equation}\label{sm:witness-bound}
w^*Bw=v^*(B-\delta)_+v,
\qquad
\|w\|^2\leq
\frac{\|A\|+\|v^*(B-\delta)_+v-A\|}{\delta}.
\end{equation}
Thus coordinate comparisons into a fixed positive cut have
bounded witnesses into the uncut target. This conclusion is
independent of matrix size.
\subsection{Fractions of an existing model}
\begin{proof}[Proof of \Cref{sm:stabilizers}\textup{(i)}]
Take c.p.c. lifts \(\chi_n:S\to Q\) of \(\chi\), using
nuclearity and the Choi--Effros lifting theorem
\cite[Theorem~3.10]{ChoiEffrosLifting}.
The constant-ideal assertion of \Cref{seq:all-traces} gives
\[
\Ideal_D(\chi(x_l))=\Ideal_D(Q(U_{x_l})).
\]
Compact support of the controller in
\eqref{sm:error-controllers}, with an extra cut before passing
to coordinates, consequently gives an integer \(M_q\) and
\(d_q>0\) such that
\begin{equation}\label{sm:model-lower}
d_\tau(c_q)
\leq M_q d_\tau((\chi_n(x_l)-d_q)_+)
\quad\text{eventually, for every }\tau.
\end{equation}
Here is the cut detail. Write
\([c_q]\leq[(z-\gamma)_+]\), with \(z\in Q(U_{x_l})_+\).
The constant \((z-\gamma/2)_+\) lies in the ideal generated
by \(\chi(x_l)\). Its suitable smaller cut is therefore
below finitely many copies of a positive cut of \(\chi(x_l)\).
Leave a further cut between \((z-\gamma)_+\) and that source
before lifting the comparison. The source perturbation lemma
then gives \eqref{sm:model-lower} at every late coordinate.
Both its multiplicity and target tolerance depend only on the
request, not on the coordinate.
For an upper test, choose a strictly positive $h_i\in Q(V_i)$.
The constant-ideal assertion gives
$\chi(a_i)\in\Ideal_D(Q(V_i))=\Ideal_D(h_i)$.
Compact containment between two positive cuts of $\chi(a_i)$
therefore gives a finite integer $N_i$ and $\kappa_i>0$ with
\[
[(\chi(a_i)-\eps_i/2)_+]\leq N_i[(h_i-\kappa_i)_+].
\]
Let $c_i\in Q(V_i)_+$ be the folded sum of these $N_i$ cuts.
It is compactly Cuntz supported in $Q(V_i)$.
The common-controller lemma then gives $[c_i]\leq L_i[b_i]$
for a finite integer $L_i$. Lift a comparison for the less deeply
cut source with error smaller than the remaining source slack.
The positive perturbation lemma gives
\begin{equation}\label{sm:model-upper}
d_\tau((\chi_n(a_i)-\eps_i)_+)
\leq L_i d_\tau(b_i)
\quad\text{eventually, for every }\tau.
\end{equation}
Thus both the controller $b_i$ and its multiplicity are fixed
before the UHF fractions are selected.
At stage \(j\), choose a positive rational \(t_j\) with
\[
t_j\leq j^{-1},
\qquad t_jL_i\leq j^{-1}\quad(i\leq j).
\]
Only after choosing \(t_j\), choose an increasing sequence
of coordinate thresholds \(N_j\) so that all the first
\(j\) coordinate comparisons hold and
\begin{equation}\label{sm:model-delay}
M_qr_q(n)\leq t_j
\qquad(q\leq j, n\geq N_j).
\end{equation}
For \(N_j\leq n<N_{j+1}\), let \(p_n\) be a projection
of normalized trace \(t_j\) in an extra UHF factor and use
\[
\theta_n(a)=\chi_n(a)\otimes p_n.
\]
After the fixed trace-preserving identifications these are
c.p.c. maps into \(Q\). Their multiplicative defects tend
to zero because those of \(\chi_n\) do, so they define a
homomorphism \(\theta:S\to D\).
Equation~\eqref{sm:model-upper} gives
\eqref{sm:upper-goal} with \(s_{i,n}\leq j^{-1}\) on the
\(j\)-th stage. Moreover,
\begin{align}
d_\tau(A_{q,n})
&\leq r_q(n)d_\tau(c_q)\notag\\
&\leq M_qr_q(n)d_\tau((\chi_n(x_l)-d_q)_+)
\leq d_\tau((\theta_n(x_l)-d_q)_+).
\label{sm:fraction-lower}
\end{align}
All inequalities use extended nonnegative arithmetic;
the scalar factors are strictly positive. Pointwise order
in \(\Cu(Q)\) gives the coordinate comparison. Apply
\eqref{sm:witness-bound} to the target
\((\theta_n(x_l)-d_q/2)_+\), cut further at \(d_q/2\).
Bounded witnesses now pass to \(D\), proving
\eqref{sm:fixed-lower-cut} for the enumerated tolerances,
and hence for every tolerance.
It remains to check \eqref{sm:coarse-fullness}. If
\(y\in S(U_x)_+\), then
\(\chi(y)\in\Ideal_D(\chi(x))\). For every source
tolerance, a slightly larger source cut is below finitely
many copies of a fixed cut of \(\chi(x)\). Passing to
coordinates with slack gives this rank inequality eventually.
Tensoring both sides with the same \(p_n\) multiplies both
ranks by its positive trace, so the inequality remains true.
Pointwise coordinate order and
\eqref{sm:witness-bound} give
\((\theta(y)-\eps)_+\in\Ideal_D(\theta(x))\).
Letting \(\eps\downarrow0\), then using linearity and
closure, proves \eqref{sm:coarse-fullness}.
Finally \(Nt_j\leq1\) eventually for every fixed \(N\),
so \(N\) equal rational subprojections fit in the original
UHF unit. They give the stated copies inside the spare model.
\end{proof}
\subsection{Cone blocks with a fixed lower signal}
Without a model to cut down, we must first build finite matrix maps
whose values detect each requested controller. The construction has
three ingredients: a fixed quotient-norm threshold for the controlling
input, finitely many positive support elements with plateaux, and
finite-dimensional approximations of the corresponding quotient cones.
The plateaux convert a matrix norm lower bound into a Cuntz comparison
at the fixed threshold. Stage-dependent matrix sizes will change only
the rank multiplicities.
We use the following finite approximation form of cone
quasidiagonality: for a separable \(C^*\)-algebra \(R_0\), a finite
subset of \(C_0((0,1])\otimes R_0\), and a positive tolerance,
there is a c.p.c. map to a matrix algebra which is
multiplicative to that tolerance and preserves the norms of
the specified elements to that tolerance. This is
Voiculescu's cone theorem \cite{Voiculescu}; a direct proof in precisely this
form is \cite[Proposition~33]{Shulman}.
The ideal-covering strategy in the finite cone stage is inspired by
Bosa, Gabe, Sims, and White; compare \cite[Lemma~3.5]{BGSW}.
Their target is \(\mathcal O_\infty\)-stable. The fixed-cut estimates
and UHF trace-small controls for the present target are proved here,
not imported from that lemma.
\begin{lemma}[A finite cone stage]\label{sm:cone-stage}
Let \(S=C_0((0,1])\otimes P\). Fix finitely many upper
tests \eqref{sm:upper-tests}, finitely many lower requests
\eqref{sm:error-controllers}, and a finite multiplicativity
test set. For each lower request one can fix a number
\(h_q>0\), independently of all later stages, such that the
following holds.
For arbitrarily small multiplicativity error there is a
c.p.c. map \(\Phi:S\to M_k(Q)\), with finite constants
\(L_i\) and integers \(M_q\geq1\), satisfying
\begin{equation}\label{sm:stage-bounds}
\begin{split}
d_\tau(\Phi(a_i))&\leq L_i d_\tau(b_i),\\
d_\tau(c_q)&\leq
M_q d_\tau((\Phi(x_l)-4h_q)_+)
\qquad(q=(l,\eta)),
\end{split}
\end{equation}
for every \(\tau\in T(Q)\), with the unnormalized matrix
extension of \(\tau\).
\end{lemma}
\begin{proof}
We first choose the lower threshold, once for each request.
Write
\[
[c_q]\leq[(z_q-\gamma_q)_+],
\qquad z_q\in Q(U_{x_l})_+,\quad\gamma_q>0.
\]
For \(z\in X\), denote by \((x_l)_z\) the image in
\(C_0((0,1])\otimes(P/P_z)\), where \(P_z\) is the
primitive ideal labeled by \(z\). The sets
\[
O_q(h)=\{z:\|(x_l)_z\|>8h\}
\]
are open and increase to \(U_{x_l}\) as \(h\downarrow0\).
Approximation in this increasing union of ideals allows us
to choose \(h_q>0\) such that
\begin{equation}\label{sm:envelope}
z'_q=(z_q-\gamma_q/2)_+\in Q(O_q(h_q)),
\qquad [c_q]\ll[z'_q].
\end{equation}
Keep \(h_q\) fixed from now on.
For a point \(z\in O_q(h_q)\), choose an open neighborhood
\[
N_z\subseteq O_q(h_q)\cap
\bigcap_{\{i:(a_i)_z\ne0\}}\supp_X(b_i).
\]
It exists because \((a_i)_z\ne0\) implies \(z\in V_i\),
and \(\supp_X(b_i)\) contains \(V_i\).
Functional calculus on a positive element of \(Q(N_z)\)
nonzero at \(z\) gives positive contractions \(h_z,k_z\)
with
\begin{equation}\label{sm:plateau}
h_zk_z=k_z,
\qquad (k_z)_z\ne0,
\qquad h_z\text{ compactly Cuntz-supported in }Q(N_z).
\end{equation}
For example, first take a nonzero positive cut, make \(h_z\)
equal to one on a smaller spectral region, and support
\(k_z\) inside that region.
The ideal supports of all these \(k_z\)'s cover
\(O_q(h_q)\). By \eqref{sm:envelope}, finitely many of
them suffice for
\begin{equation}\label{sm:finite-cover}
[c_q]\leq M_q\sum_{t=1}^{\ell_q}[k_{qt}]
\end{equation}
for some integer \(M_q\).
This uses compact containment in the Cuntz semigroup: the
ideal generated by all the \(k_z\)'s contains \(z'_q\),
and \([c_q]\ll[z'_q]\) is captured by a finite sum with
finite multiplicity. No compactness of the whole open set
\(O_q(h_q)\) is assumed.
For each selected center \(z\), put
\(S_z=C_0((0,1])\otimes(P/P_z)\), and let
\(\alpha_t\), \(0\leq t\leq1\), be cone scaling:
\[
(\alpha_t f)(r)=f(tr),
\qquad \alpha_0=0.
\]
The scaled images of a finite test set form norm-compact
families. Choose finite nets in those families and their
products. Cone quasidiagonality, followed by contractivity
to pass from the nets to the families, gives a c.p.c. map
\(\phi_z:S_z\to M_{d_z}\) with the required uniform
multiplicativity error and
\begin{equation}\label{sm:matrix-signal}
\|\phi_z((x_l)_z)\|>7h_q.
\end{equation}
Define a matrix-valued cone function by
\[
t\longmapsto\phi_z(\alpha_t(s_z)).
\]
It is norm-continuous and zero at \(t=0\). Evaluate its
entries at \(h_z\). The resulting map
\(\psi_z:S\to M_{d_z}(Q)\) is c.p.c.; it has the same
multiplicativity bound because the evaluation map is a
homomorphism. Its range lies in \(M_{d_z}(\Her(h_z))\).
For the lower estimate, choose a unit vector \(v\in\C^{d_z}\)
which is an eigenvector of the positive matrix
\(\phi_z((x_l)_z)\) for an eigenvalue \(\lambda>7h_q\).
Equation~\eqref{sm:plateau} implies that evaluation on
\(k_z^{1/2}\) is evaluation at parameter one. Consequently
the rectangular column
\[
w=(\lambda-4h_q)^{-1/2}\,v\otimes k_z^{1/2}
\]
satisfies
\begin{equation}\label{sm:plateau-witness}
w^*(\psi_z(x_l)-4h_q)_+w=k_z,
\qquad \|w\|\leq(3h_q)^{-1/2}.
\end{equation}
Combining these comparisons with
\eqref{sm:finite-cover} gives the second inequality of
\eqref{sm:stage-bounds} for the direct sum of all the
selected blocks.
For an upper test, a block is zero if \((a_i)_z=0\).
Otherwise \(h_z\) is compactly Cuntz-supported in an ideal
contained in \(\Ideal_Q(b_i)\). It is therefore below
finitely many copies of \(b_i\). Since the block range
is supported by \(1_{d_z}\otimes h_z\), its rank is at
most \(d_zd_\tau(h_z)\). Summing these finite bounds gives
the first inequality of \eqref{sm:stage-bounds}.
The direct sum is contractive, since its norm is the maximum
of the block norms. Its internal matrix trace is the sum
of the block traces. This proves the lemma.
\end{proof}
The finite stage now has a lower spectral cut independent of the
accuracy. To turn these maps into a locally small homomorphism, we
choose their UHF fractions next and delay the coordinates until the
prescribed errors are smaller than those fractions.
\begin{proof}[Proof of \Cref{sm:stabilizers}\textup{(ii)}]
At stage \(j\), apply \Cref{sm:cone-stage} to the first
\(j\) upper tests, lower requests, and an increasing dense
list of algebra tests, with multiplicativity error at most
\(j^{-1}\). Write the resulting map as \(\Phi_j\),
and its constants as \(L_{ij}\) and \(M_{qj}\).
Choose a positive rational \(t_j\) satisfying
\begin{equation}\label{sm:cone-fraction}
t_j\leq j^{-1},
\qquad t_jL_{ij}\leq j^{-1}\quad(i\leq j),
\qquad \beta_{qj}:=t_j/M_{qj}>0.
\end{equation}
After all the stage maps and fractions have been chosen,
choose increasing coordinate thresholds \(N_j\) with
\begin{equation}\label{sm:cone-delay}
r_q(n)\leq\beta_{qj}
\qquad(q\leq j, n\geq N_j).
\end{equation}
On \(N_j\leq n<N_{j+1}\), tensor \(\Phi_j\) with a UHF
projection of trace \(t_j\), and fold its finite matrices
to obtain \(\theta_n:S\to Q\).
These maps are c.p.c. and asymptotically multiplicative.
Their quotient is therefore a homomorphism \(\theta\).
For a fixed upper test, \eqref{sm:stage-bounds} and
\eqref{sm:cone-fraction} give
\eqref{sm:upper-goal} eventually with bound \(j^{-1}\).
Thus \(\theta(S(U))\subseteq J(U)\).
For a lower request \(q=(l,\eta)\),
\begin{equation}\label{sm:cone-coordinate-comparison}
d_\tau(A_{q,n})
\leq r_q(n)d_\tau(c_q)
\leq d_\tau((\theta_n(x_l)-4h_q)_+)
\end{equation}
eventually at every coordinate. Pointwise trace order in
\(\Cu(Q)\) converts this into Cuntz comparison.
Put \(B_{q,n}=(\theta_n(x_l)-2h_q)_+\), choose comparison
witnesses with error at most \(n^{-1}\), and apply
\eqref{sm:witness-bound} with tolerance \(2h_q\).
The resulting witnesses have squared norms at most
\[
\frac{\sup_n\|A_{q,n}\|+1}{2h_q},
\]
independently of the stage, its centers, and its matrix size.
They prove
\[
[(|e_l|-\eta)_+]
\leq[(\theta(x_l)-2h_q)_+]
\quad\text{in }\Cu(D).
\]
This establishes \eqref{sm:fixed-lower-cut} and then
\eqref{sm:local-and-lower}. A very small \(\beta_{qj}\)
changes only the delay \(N_j\); it never changes the fixed
cut \(h_q\).
All comparisons may be made in the common block sequence
algebra before folding. Include the finitely many prescribed
error representatives in each stage's folding tests in
\Cref{st:folding}. The coordinate conjugacies then preserve
their original quotient classes, and the norm bounds above
are unchanged. This proves the separate-corner assertion.
\end{proof}
\subsection{Coefficient families}
\begin{corollary}[Simultaneous coefficient domination]
\label{sm:coefficient-domination}
Suppose that one of the hypotheses of
\Cref{sm:stabilizers} holds. Let \(E_j\) be countably
many bounded multilinear maps
\[
E_j:S\times Z_{j,1}\times\cdots\times Z_{j,r_j}\longrightarrow D,
\]
where the parameter spaces are separable normed spaces.
Assume that
\[
E_j(S(U),Z_{j,1},\ldots,Z_{j,r_j})\subseteq J(U)
\quad\text{for every open }U.
\]
Then the stabilizer may also be chosen so that, for every
\(x\in S_+\),
\begin{equation}\label{sm:all-coefficients}
E_j(S(U_x),Z_{j,1},\ldots,Z_{j,r_j})
\subseteq\Ideal_D(\theta(x)).
\end{equation}
The same conclusion holds for norm-continuous separably
parametrized families of such maps, with local norm bounds,
and for any choice of the localized multilinear input.
\end{corollary}
\begin{proof}
Choose a countable family \(\mathcal B\subseteq S_+\) of
positive cuts which is an ideal basis: for every \(x\in S_+\),
the elements of \(\mathcal B\cap\Ideal_S(x)\) generate
\(\Ideal_S(x)\). Such a family is obtained from a countable
norm-dense positive set and rational cuts, using the
positive-element perturbation lemma. In particular,
\begin{equation}\label{sm:projected-cover}
U_x=\bigcup_{y\in\mathcal B\cap\Ideal_S(x)}U_y.
\end{equation}
For each \(y\in\mathcal B\), prescribe the errors
\(E_j(a,z_1,\ldots,z_{r_j})\), where \(a\) runs through
a countable dense set in \(S(U_y)\), and the other variables
run through countable dense sets in their parameter spaces.
These are countably many requests with controlling element
\(y\), to which \Cref{sm:stabilizers} applies.
Continuity gives the desired containment for every input in
\(S(U_y)\) and every parameter value.
Now fix \(x\). If \(y\in\mathcal B\cap\Ideal_S(x)\),
the homomorphism property gives
\(\theta(y)\in\Ideal_D(\theta(x))\).
By \eqref{sm:projected-cover}, \(S(U_x)\) is the closed sum
of the ideals \(S(U_y)\) in that equation. Linearity in
the localized input and closedness of the target ideal give
\eqref{sm:all-coefficients}. Dense parameter sets and local
norm continuity handle the asserted continuous families.
Notice that the argument uses
\(y\in\Ideal_S(x)\), not merely \(U_y\subseteq U_x\).
Thus it does not presume coarse fullness of the cone
stabilizer.
\end{proof}
\begin{remark}\label{sm:multiplier-coefficients}
Two forms of \Cref{sm:coefficient-domination} will be used.
If \(I\) is a separable error algebra and
\(d:S\to D\) is bounded linear with
\(d(S(U))I\subseteq J(U)\), apply the corollary to
\(E(a,b)=d(a)b\), \(b\in I\), and, when needed, to the
adjoint and opposite-product maps. This gives the full
projected-support bounds on the hereditary support
generators.
Alternatively, let \(H\subseteq D\) be a given
\(\sigma\)-unital hereditary algebra, possibly nonseparable,
and let \(d:S\to M(H)\) be a bounded linear map whose
localized coefficients lie in \(J(U)\).
Choose a countable approximate unit \((u_m)\) for \(H\)
and prescribe \(u_md(a)u_m\) by the corollary. For
\(h,k\in H\),
\[
h d(a)k=\lim_m h u_m d(a)u_m k.
\]
Thus all multiplier coefficients are controlled. The
countability needed here comes from \(\sigma\)-unitality,
not from separability of \(H\).
\end{remark}
==== END The-trace-cone-classifies-Razak-Jacelon-stabilizations//sections/04-small-stabilizers.tex ====
==== BEGIN The-trace-cone-classifies-Razak-Jacelon-stabilizations//sections/05-uniqueness.tex ====
\section{Norm uniqueness of sequence models}\label{sec:uniqueness}
We prove uniqueness for the models of \Cref{seq:model}, allowing any of
the spaces $Y$ and measures $m$ specified there. The uniqueness
argument uses the coarse labels
\[
S=S_Y=C_0(Y)\otimes P,\qquad S(U)=C_0(Y)\otimes P(U),
\qquad U\subseteq X\text{ open}.
\]
The same statements apply if the prescribed measure is multiplied by a
strictly positive constant. A coordinate unitary will always mean a
unitary in $\mathcal M(Q)$; convergence of coordinate conjugacies is in
norm on each fixed input.
\begin{theorem}[Norm uniqueness]\label{un:uniqueness}
Let $p,q:S_Y\to Q_\infty$ be models for the same data. For any
c.p.c. lifts $p_n,q_n:S_Y\to Q$, there are
$u_n\in\mathcal U(\mathcal M(Q))$ such that
\[
\lim_{n\to\infty}\|u_np_n(a)u_n^*-q_n(a)\|=0,
\qquad a\in S_Y.
\]
\end{theorem}
There are two steps. Equal trace data first give, after conjugacy and
rational averaging, a difference in every local error ideal $J(U)$.
A small fraction of a common spare model then absorbs that difference
in norm. Splitting each original model into two UHF halves supplies
the spare: change the first half while keeping the second, and then
change the second while keeping the first. The intermediate absorption
statements allow a general labeling space, so that the cone correction
can use them with its finer ideal labels.
\subsection{Compact sets of weights and tracial alignment}
\begin{lemma}[Controlled trace evaluations]\label{un:trace-cap}
Let $C$ be a $C^*$-algebra and $b\in(C\otimes\K)_+$. The set
\[
K_b=\{\gamma\in T(C):d_\gamma(b)\leq1\}
\]
is compact and convex. If $x\in(C\otimes\K)_+$ satisfies
$\sup_{\gamma\in K_b}d_\gamma(x)\leq L<\infty$, then
$\gamma\mapsto\gamma(x)$ is finite, continuous, and affine on $K_b$.
The constant $L$ may depend on $x$.
Consequently this applies separately to all the first and square
evaluations of a fixed completely positive candidate whose tested
values have bounded rank on $K_b$. It also applies to
\[
\gamma\longmapsto
\gamma\bigl(\lvert u x_1u^*-x_2\rvert^2\bigr),
\qquad u\in\mathcal U(C^{\sim}),
\]
whenever the positive elements $x_1,x_2$ have uniformly bounded ranks
there, with a rank bound independent of $u$.
\end{lemma}
\begin{proof}
Rank evaluation is lower semicontinuous on the compact trace cone, so
$K_b$ is closed. Its convexity follows from additivity and positive
homogeneity of rank. For $\eta>0$ the assumed rank bound gives
\begin{equation}\label{un:cut-loss}
0\leq\gamma(x)-\gamma((x-\eta)_+)\leq\eta L,
\qquad \gamma\in K_b,
\end{equation}
and $\gamma(x)\leq\|x\|L$. If $\gamma_i\to\gamma$ in $K_b$, the
defining cut inequalities for the trace topology imply
\[
\gamma(x)\leq\liminf_i\gamma_i(x)
\leq\limsup_i\gamma_i(x)
\leq\gamma(x)+\eta L.
\]
Letting $\eta\downarrow0$ proves continuity. Affineness follows from
the cone operations; it does not require cancellation in that cone.
Squaring a positive element does not change its support rank. Finally,
support subadditivity and unitary invariance give
\[
d_\gamma\bigl(\lvert u x_1u^*-x_2\rvert^2\bigr)
\leq d_\gamma(x_1)+d_\gamma(x_2),
\]
which proves the last assertion.
\end{proof}
\begin{lemma}[Barycenters on a trace cap]\label{un:barycenter}
Let $K_b$ be as in \Cref{un:trace-cap}. For every probability measure
$\mu$ on $K_b$ there is $\bar\gamma\in K_b$ such that
\begin{equation}\label{un:barycenter-formula}
v(\bar\gamma)=\int_{K_b}v\,d\mu
\end{equation}
for every finite continuous affine real function $v$ on $K_b$.
For a positive measure of finite mass $M>0$, its integral is instead
$M v(\bar\gamma)$, with $\bar\gamma$ chosen for $\mu/M$; zero measures
contribute zero. Signed measures may be treated by their two positive
parts.
\end{lemma}
\begin{proof}
Approximate $\mu$ weakly by finitely supported probability measures
$\mu_j=\sum_k t_{jk}\delta_{\gamma_{jk}}$, omitting terms of weight
zero. Their finite barycenters
$\beta_j=\sum_k t_{jk}\gamma_{jk}$ belong to $K_b$. Take a subnet
converging in the compact set $K_b$ to $\bar\gamma$. For every $v$ in
the statement,
\[
v(\bar\gamma)=\lim_jv(\beta_j)
=\lim_j\int v\,d\mu_j=\int v\,d\mu.
\]
The same subnet works for all these functions. Neither uniqueness of
$\bar\gamma$ on discontinuous evaluations nor subtraction of extended
weights is asserted or used.
\end{proof}
\Needspace{3\baselineskip}
\begin{lemma}[Alignment for finitely many weights]\label{un:finite-traces}
Let $C$ be separable, let $S$ be separable and nuclear, and let
$p,q:S\to C$ be homomorphisms with equal pullbacks on $T(C)$.
For a finite list of pairs $(s_i,\gamma_i)$, where $s_i\in S_+$ and
\[
d_{\gamma_i}(p(s_i))+d_{\gamma_i}(q(s_i))<\infty,
\]
and any $\delta>0$, there is $u\in\mathcal U(M_2(C)^{\sim})$ such that
\begin{equation}\label{un:finite-L2}
(\gamma_i\otimes\Tr_2)
\bigl(\lvert u(p(s_i)\oplus0)u^*-(q(s_i)\oplus0)\rvert^2\bigr)<\delta
\quad\text{for every }i.
\end{equation}
The added matrix direction is internal and has unnormalized trace.
\end{lemma}
\begin{proof}
The equality of pullback weights implies equality of all their ranks.
By \cite[Lemma~5.1]{CGNN}, the support projections of $p(s)$ and $q(s)$
are equivalent in $C^{**}$ for every $s\in S_+$. Their images remain
equivalent under the normal extension of every representation of $C$.
The finite-rank assumption puts $p(s_i)$ and $q(s_i)$ in
$\Fin(\gamma_i)$. After zero padding, $M_2(\Fin(\gamma_i))$ is invariant under
$M_2(C)^{\sim}$-unitaries, so the conjugates and squared differences
below remain in that finite ideal.
For each selected weight, take its trace representation $\pi_i$ on
its densely finite ideal after quotienting by its zero ideal, and
extend it nondegenerately to $C$. Let $\Tr_i$ be the resulting normal
semifinite trace on $M_i=\pi_i(C)''$. Form
$\pi=\pi_0\oplus\pi_1\oplus\cdots\oplus\pi_r$, with $\pi_0$
a faithful representation on a separable Hilbert space, and put
$M=\pi(C)''$. The trace representations can also be chosen separably
acting: their densely finite ideals are separable and have countable
approximate units of finite-trace cuts. Thus $M$ has separable predual.
Restriction to the $i$th reducing Hilbert summand gives a normal
surjective homomorphism $q_i:M\to M_i$. Its kernel is
$M(1-z_i)$ for a central projection $z_i$, and $q_i$ identifies
$Mz_i$ with $M_i$. Define
\[
\widehat\gamma_i=\Tr_i\circ q_i.
\]
This is the transported normal semifinite trace on $Mz_i$, extended
by zero on $M(1-z_i)$. The projections $z_i$ may overlap; the
construction is made separately for each weight. On the densely
finite ideal of $\gamma_i$ one has
$\widehat\gamma_i\pi=\gamma_i$, so it computes all designated
finite-rank tests and their squares.
We apply von Neumann uniqueness only after a common zero padding and
unitization. Let $S^\dagger$ be the forced unitization, and define
unital homomorphisms into $M_2(M)$ by
\[
\widehat p(a+\lambda1)=(\pi p(a)\oplus0)+\lambda1_{M_2(M)},\qquad
\widehat q(a+\lambda1)=(\pi q(a)\oplus0)+\lambda1_{M_2(M)}.
\]
These maps have equal support ranks on every positive element of
$S^\dagger$. If its scalar part is zero, this follows from the
support equivalences already proved. If the scalar part is positive,
then on the properly infinite central part of $M$ both support
projections contain the second matrix corner, which is equivalent to
the unit of $M_2(M)$ there. On the finite central part, every normal
finite trace has equal pullbacks along $\pi p$ and $\pi q$; this follows
by restricting it to $C$ and using the hypothesis. The unital
extensions also have equal pullbacks, hence equal ranks by monotone
spectral calculus. Normal traces determine equivalence of projections
in a finite von Neumann algebra. This proves the claim on both parts.
The source $S^\dagger$ is nuclear. We use the unital form of von Neumann
uniqueness in \cite[Proposition~2.1, Corollary~3.10, and Section~4]{CGNN}.
For completeness, the properly infinite factor argument needs only the
partial-isometry approximation of \cite[Lemma~3.6]{CGNN} and the following
unital completion. Including $1$ among the inputs gives partial
isometries $v$ whose initial projections $e=v^*v$ tend strongly to $1$.
For finitely many normal states choose $e_0\leq e$ arbitrarily close to
$e$ on those states, with
$1-e_0\sim1-ve_0v^*\sim1$: take $e_0=e$ if $e$ is finite, and otherwise
remove an infinite tail decreasing strongly to zero. Complete $ve_0$
to a unitary $u$. Since $(u-v)e_0=0$, Cauchy--Schwarz gives
\[
|\rho(u^*xu-v^*xv)|\leq4\|x\|\rho(1-e_0)^{1/2}.
\]
Thus the partial-isometry conjugacies can be replaced by unitary
conjugacies in the weak star topology; \cite[Lemma~3.9]{CGNN} makes
them strong star. Section~4 then assembles these unital factor conclusions by
central decomposition; separability lets one impose the support
comparisons on one countable dense set of positive cuts before
selecting the factorwise unitaries. We obtain $w_j\in M_2(M)$ with
\[
w_j(\pi p(a)\oplus0)w_j^*\longrightarrow\pi q(a)\oplus0
\quad\text{strong star},\qquad a\in S.
\]
The unital extensions are needed here: support ranks of nonunital maps
alone do not compare the complements of their support projections.
For clarity, the passage to $L^2$ uses more than strong convergence.
Put $x_j=w_j(\pi p(s_i)\oplus0)w_j^*$ and
$x=\pi q(s_i)\oplus0$. Write
$T_i=\widehat\gamma_i\otimes\Tr_2$. Their $T_i$-norms are finite and satisfy
\[
\|x_j\|_{2,T_i}^2
=\gamma_i(p(s_i^2))
=\gamma_i(q(s_i^2))
=\|x\|_{2,T_i}^2.
\]
For a bounded test $y\in L^1(T_i)\cap
L^2(T_i)$ supported on trace-finite projections, its
pairing with $x_j-x$ is the normal bounded functional
$t\mapsto T_i(y^*t)$ and tends to zero. Such tests
are dense in $L^2(T_i)$; the common
$L^2$ bound therefore gives weak $L^2$ convergence. Equality of norms
now gives $\|x_j-x\|_{2,T_i}\to0$.
Choose one $w_j$ meeting the finite list to sufficiently small
tolerance. A bounded self-adjoint logarithm of this unitary can be
approximated strong star, within its norm bound, by self-adjoint
elements of $M_2(\pi(C))^{\sim}$, by Kaplansky density
\cite[Theorem~1]{KaplanskyDensity}. Exponentiation
gives unitaries from $M_2(C)^{\sim}$ converging strong star to $w_j$.
Their conjugates again have the same designated $L^2$ norms, so the
preceding finite-trace-test argument gives $L^2$ convergence for this
approximation too. On these conjugates the normal traces compute
the original weights, since their finite ideals are invariant under
$M_2(C)^{\sim}$-unitaries. This proves \eqref{un:finite-L2}.
\end{proof}
\begin{lemma}[Tracial alignment]\label{un:alignment}
If $p,q:S\to D$ are models with the same data, then rational diagonal
averaging, coordinate multiplier conjugacy, and the foldings of
\Cref{st:folding} give models $\widetilde p,\widetilde q:S\to D$ with
\begin{equation}\label{un:local-difference}
(\widetilde p-\widetilde q)(S(U))\subseteq J(U)
\qquad(U\subseteq X\text{ open}).
\end{equation}
Each new model is coordinate approximately unitarily equivalent to
its respective original model. The operations may be performed in
one changing block while a prescribed spare block is carried along
unchanged, apart from its common amplification and folding.
\end{lemma}
\begin{proof}
By \Cref{seq:all-traces}, choose a separable subalgebra $D_0$ containing
the two ranges, the required constant representatives, and the
comparison witnesses, so that $p$ and $q$ agree on all traces of
$D_0$. Work in a common stabilized corner when needed.
Choose a countable local positive test family $s_i$, dense in linear
span in each of the basis ideals used in \Cref{seq:diagonal}.
Each test is cut-supported in both factors and has compact support
in $Y$. Choose an enlarged elementary test $\bar s_i$ and a fixed
$\kappa_i>0$ with $s_i\precsim(\bar s_i-\kappa_i)_+$, using the
positive cut in each factor. Hence
$p(s_i)\precsim(p(\bar s_i)-\kappa_i)_+$, and the same holds for $q$.
The model-independent upper sandwich, applied to this fixed cut of
$\bar s_i$, supplies a common compact-Cuntz-supported constant;
dominating it in the chosen containing open gives
$b_i\in Q(U_i)_+$ and a finite constant $L_i$ such that
\begin{equation}\label{un:controller-rank}
d_\gamma(p(s_i))+d_\gamma(q(s_i))
\leq L_i d_\gamma(b_i),\qquad\gamma\in T(D_0),
\end{equation}
for a finite $L_i$. The constants and the comparisons are included
in $D_0$. One can retain $b_i$ whenever later finite lists grow.
Fix a finite initial list and $\delta>0$. For
$u\in\mathcal U(M_2(D_0)^{\sim})$, define the nonnegative continuous
functions
\[
f_{u,i}(\gamma)
=(\gamma\otimes\Tr_2)
\bigl(\lvert u(p(s_i)\oplus0)u^*-(q(s_i)\oplus0)\rvert^2\bigr),
\qquad \gamma\in K_{b_i}.
\]
Here $\Tr_2$ is unnormalized: the common zero padding leaves
\eqref{un:controller-rank} unchanged. Continuity follows from
\Cref{un:trace-cap}. We claim that zero lies
in the closed convex hull of the tuples $(f_{u,i})_i$ in
$\bigoplus_i C(K_{b_i},\R)$. Otherwise a separating real functional
would be bounded below by a positive constant on all these tuples.
Represent its components by signed measures. Because $f_{u,i}\geq0$,
replacing each measure by its positive part retains that lower bound.
By \Cref{un:barycenter}, the resulting functional on these functions
is a finite sum of positive multiples of evaluations at individual
weights in the corresponding caps. \Cref{un:finite-traces} makes
all these evaluations simultaneously arbitrarily small, a
contradiction.
It follows that a finite convex combination satisfies
\begin{equation}\label{un:uniform-square}
\sup_{\gamma\in K_{b_i}}
\sum_{k=1}^r t_k f_{u_k,i}(\gamma)<\delta
\qquad\text{for every tested }i.
\end{equation}
The coefficients may be chosen positive and rational: the finite
functions being averaged are bounded, so sufficiently small changes
of the coefficients preserve the strict inequality. Write
$t_k=m_k/l$. Before folding, work in $B_l=M_l(M_2(Q))$ with trace
$\tau\otimes\Tr_2\otimes\operatorname{tr}_l$: only this last,
additional averaging direction is normalized. The two averaged maps
are diagonal copies of $\Ad(u_k)(p\oplus0)$ and of $q\oplus0$,
with $m_k$ occurrences of the $k$th block. The original algebra $D_0$
has the fixed corner embedding
\[
\iota_l:D_0\longrightarrow(B_l)_\infty,
\qquad x\longmapsto(x\oplus0)\otimes1_l.
\]
We record explicitly the resulting coordinate estimate. Lift the
finitely many unitaries to unitary multiplier sequences, and use
c.p.c. representatives of $p,q$. Write $z_{i,n}$ for the difference
of their averaged values, after folding into $Q$. For every fixed
$t>0$, we first prove the absolute estimate
\begin{equation}\label{un:cap-coordinate-rank}
d_\tau((|z_{i,n}|-t)_+)
\leq \frac{4\delta}{t^2}
\qquad\bigl(d_\tau(b_i)\leq1\bigr)
\end{equation}
for every sufficiently large $n$, uniformly over this cap. Suppose
that this absolute bound fails arbitrarily late, and choose
violating coordinates and weights $\tau_n$. The trace identities
of \Cref{st:folding} pull these weights back, before folding, to
$\tau_n\otimes\Tr_2\otimes\operatorname{tr}_l$. The subsequent multiplier
conjugations preserve these identities. Positive cuts commute with
the coordinate folding maps, so regularization commutes with
them as well. Let $\rho_l$ be the regularized trace before folding
and put $\gamma=\rho_l\circ\iota_l$. It is a trace on the same
original $D_0$, and \Cref{seq:regularized} gives
\[
d_\gamma(b_i)
=\sup_{\eta>0}\lim_\omega d_{\tau_n}((b_i-\eta)_+)\leq1.
\]
Thus $\gamma\in K_{b_i}$. Put
$e_{k,i}=u_k(p(s_i)\oplus0)u_k^*-(q(s_i)\oplus0)\in M_2(D_0)$. If $z_i$ denotes the
unfolded diagonal difference, matrix-unit traciality and finite
additivity give
\begin{equation}\label{un:averaged-limit}
\rho_l(z_i^*z_i)
=\sum_k\frac{m_k}{l}(\gamma\otimes\Tr_2)(e_{k,i}^*e_{k,i})<\delta.
\end{equation}
Indeed, restriction to the first internal corner gives $\gamma$,
so the matrix extension on $M_2(D_0)$ is $\gamma\otimes\Tr_2$.
For every positive $a\in M_2(D_0)$,
$\rho_l(a\otimes e_{jj})=(\gamma\otimes\Tr_2)(a)/l$: the $l$
averaging corners are equivalent and sum to $a\otimes1_l$. This uses only finite positive sums, so also holds
when a value is infinite. It follows that
\eqref{un:averaged-limit} is valid without a bounded-weight
assumption. On the other hand, functional calculus gives
\[
\tau\bigl((|z_{i,n}|^2-t^2/2)_+\bigr)
\geq \frac{t^2}{2}d_\tau((|z_{i,n}|-t)_+).
\]
Regularization and the violations of the absolute estimate therefore
give a square trace at least $2\delta$, contrary to
\eqref{un:averaged-limit}. This proves
\eqref{un:cap-coordinate-rank}. If $0<d_\tau(b_i)<\infty$, apply
that estimate to $\tau/d_\tau(b_i)$. If $d_\tau(b_i)=0$, apply it
to every positive multiple of $\tau$, forcing the left side to be
zero. For $d_\tau(b_i)=\infty$ the desired proportional bound is
automatic. Thus, at the same sufficiently late coordinates,
\begin{equation}\label{un:coordinate-rank}
d_\tau((|z_{i,n}|-t)_+)
\leq \frac{4\delta}{t^2}d_\tau(b_i)
\qquad\text{for all }\tau\in T(Q).
\end{equation}
At stage $j$ use the first $j$ local tests, the cut tolerances
$1,1/2,\ldots,1/j$, and choose
$\delta_j<2^{-j}/(4j^2)$. Then advance far enough in that row for
\eqref{un:coordinate-rank}, the chosen algebraic tests, and their
norm bounds all to hold. Pass through the rows slowly as in
\Cref{seq:diagonal}. The averaging matrix size is fixed while
each row's uniform coordinate estimate is proved. For a fixed local
test and any fixed cut
tolerance, this gives its $J(U_i)$ estimate against the fixed $b_i$.
Closure and the local density of the tests give
\eqref{un:local-difference} for all open ideals.
For completeness, these operations preserve the asserted relation
to the original maps. At each coordinate, a diagonal sum of
conjugates of a map is conjugate, by the diagonal of the same
unitaries, to its tensor-first copy in the averaging matrix factor.
The coordinate folding can be chosen, using \Cref{st:folding}, to
carry that copy within the stage tolerance of the original on every
current input. The first-corner clause of \Cref{st:folding} removes
the common internal zero padding at the same time. Make that tolerance tend to zero during the same
diagonal choice. Inner conjugacy preserves every coordinate weight,
and vanishing norm changes preserve regularized values by cuts, so
the models keep their data. The argument applies to a designated
block of a direct sum, amplifying the spare by the same averaging
factor. Its unused part can always be placed in a separate stable
corner before folding.
\end{proof}
\subsection{Ideal-related absorption and compact perturbations}
Tracial alignment has placed the difference of two models in each
local error ideal. To remove that difference in norm, we will dominate
its coefficients by a small stabilizer and apply stable uniqueness
on a hereditary support. We first state the absorption and compact
perturbation results used in that step.
We specify the version of the ideal-related machinery used below.
An $X$-structure on a coefficient algebra $H$ is an order-preserving
assignment $U\mapsto H(U)$ of closed ideals. We do not assume that
this assignment preserves intersections. Set
\[
E_H=\ell^2(\N)\otimes H,\qquad
\mathcal K(E_H)=H\otimes\K,\qquad
\mathcal K(E_H)(U)=H(U)\otimes\K.
\]
A representation $\phi:S\to\mathcal L(E_H)$ is weakly equivariant
if
\begin{equation}\label{un:weak-equivariance}
\langle\xi,\phi(a)\eta\rangle\in H(U)
\quad(a\in S(U),\ \xi,\eta\in E_H).
\end{equation}
Equivalently, $\phi(S(U))E_H\subseteq E_HH(U)$. Every adjointable
module operator preserves these submodules. The diagonal coefficient
maps $a\mapsto\langle\xi,\phi(a)\xi\rangle$ are completely positive.
For a nuclear $S$ they and their induced quotient maps are nuclear,
since ideals and quotients of a nuclear $C^*$-algebra are nuclear
\cite[Corollary~4]{ChoiEffrosInjectivity}; other coefficients follow
by polarization. Thus weak equivariance here
implies weak $X$-nuclearity, also for homotopies.
\Needspace{3\baselineskip}
\begin{theorem}[Ideal-related absorption]\label{un:absorption}
Let $S$ be separable, exact, and lower semicontinuous over $X$, and
let $H$ be sigma-unital. Suppose
$\theta:S\to H\otimes\K$ is a nuclear $X$-full homomorphism, meaning
\[
\Ideal_{H\otimes\K}(\theta(a))=H(U_a)\otimes\K
\qquad(a\in S_+).
\]
Then its infinite repeat is weakly $X$-nuclear and absorbs every
weakly $X$-nuclear representation on the standard module, by unitary
equivalences with compact errors tending to zero on each input.
\end{theorem}
This is \cite[Theorem~11.14]{Gabe}, applied to $H\otimes\K$.
Neither separability of $H$ nor an absorption hypothesis on $H$ is
required. The repeat is taken on the stabilized module. In particular,
if a map $\theta_0:S\to H$ is first placed in the compact corner
$\theta_0\otimes e_{11}$, its stable repeat retains the other, zero
module slots. After a permutation it has the form
$\operatorname{diag}(\theta_0,0,\theta_0,0,\ldots)$.
These zero slots remain available when absorbing a degenerate
representation.
For weakly equivariant representations $\phi,\psi:S\to
\mathcal L(E_H)$ with compact difference, write $[\phi,\psi]_X$ for
the class of the graded cycle
\begin{equation}\label{un:pair-cycle}
\left(E_H\oplus E_H,
\begin{pmatrix}\phi&0\\0&\psi\end{pmatrix},
\begin{pmatrix}0&1\\1&0\end{pmatrix}\right).
\end{equation}
The two summands have opposite gradings. The compact-difference
condition supplies the compact commutators in this cycle. This
notation refers to the pair, without assigning separate classes to
the multiplier-valued maps.
\begin{proposition}[Multiplier stable uniqueness]\label{un:stable-uniqueness}
Suppose $S$ is separable, nuclear, and lower semicontinuous over $X$,
$H$ is sigma-unital, and $\theta:S\to H\otimes\K$ is an $X$-full
homomorphism.
Let $\phi,\psi:S\to\mathcal L(E_H)$ be weakly equivariant
representations with compact difference and $[\phi,\psi]_X=0$.
Writing $\theta^\infty$ for an infinite repeat, for every finite
$\mathcal F\subset S$ and $\eps>0$ there is a unitary
$v\in1+\mathcal K(E_H\oplus E_H)$ such that
\begin{equation}\label{un:proper-uniqueness}
\|v(\phi\oplus\theta^\infty)(a)v^*
-(\psi\oplus\theta^\infty)(a)\|<\eps,
\qquad a\in\mathcal F.
\end{equation}
\end{proposition}
\begin{proof}
The multiplier-pair theorem \cite[Theorem~13.9]{Gabe} supplies a
weakly $X$-nuclear representation $\Theta$ and an identity-plus-compact
unitary $u$ comparing $\phi\oplus\Theta$ with $\psi\oplus\Theta$
to any prescribed finite-test tolerance $\delta$. Here the pair
class may be read in the nuclear theory, since $S$ is nuclear.
Put $\sigma=\Theta\oplus\theta^\infty$. By
\Cref{un:absorption}, there is a module unitary $V$ from the repeat
module to the module of $\sigma$ such that
\[
\|V^*\sigma(a)V-\theta^\infty(a)\|<\eta,
\qquad a\in\mathcal F.
\]
With $W_0=1\oplus V$, set
\[
v=W_0^*(u\oplus1)W_0.
\]
It is a unitary, and $v-1$ is compact because conjugating a compact
module operator by a module unitary preserves compactness. The
error in \eqref{un:proper-uniqueness} is at most $\delta+2\eta$.
Choose these parameters with $\delta+2\eta<\eps$.
This is the finite-test common-stabilizer transfer underlying
\cite[Lemmas~2.4 and~3.4]{DadarlatEilers}; no initial value of a
continuous unitary path is needed here.
\end{proof}
\begin{lemma}[The trivially labeled factor]\label{un:KK-nullity}
For the coarse $X$-structure on $S=C_0(Y)\otimes P$ and every
sigma-unital coefficient $X$-algebra $H$,
\[
\KK(X;S,H)=0.
\]
In particular every weakly equivariant multiplier pair with compact
difference has zero class in \Cref{un:stable-uniqueness}.
\end{lemma}
\begin{proof}
Since $\W$ is simple and nuclear, every ideal of its spatial tensor
product with another $C^*$-algebra is a tensor-product ideal
\cite[Theorem~3.1]{GuptaJain}. Together with the transported labels of
\Cref{st:ideals}, this gives, after reordering the tensor factors, an
$X$-algebra $C=C_0(Y)\otimes A\otimes\K$ such that
\[
S=\W\otimes C,\qquad S(U)=\W\otimes C(U).
\]
The identity of $\W$ has zero ordinary Kasparov class. Exterior
tensoring its stabilized cycle relation and its homotopy with $C$
retains weak equivariance. Explicitly, a typical coefficient of
the tensor representation, for $c\in C(U)$, is a finite sum
\[
\sum_{i,j}\langle\xi_i,\pi(w)\xi_j\rangle
\otimes c_i^*c c_j\ \in\ \W\otimes C(U).
\]
The same formula applies throughout the homotopy. It follows that
the identity cycle of $S$ is zero for these labels. Taking its cycle
product with an $S$--$H$ cycle preserves this stabilized null relation. The identity-product homotopy identifies the
same product with the original cycle, so that cycle is zero as well.
The cycle relation and mixed product statements used here are
\cite[Propositions~12.17 and~12.20 and Lemmas~12.18, 12.19,
and~12.23]{Gabe}.
They require the first algebra to be separable and the remaining
coefficient algebras to be sigma-unital. Thus they apply to this
$H$ even when it is nonseparable. Nuclearity of $S$ also makes the
cycles and homotopies weakly $X$-nuclear, as needed above.
\end{proof}
\begin{lemma}[Hereditary support and local multiplication]
\label{un:hereditary-support}
Let $I\subseteq C$ be a separable $C^*$-subalgebra with strictly
positive element $e$, and let $k\in C_+$. Set
\[
H=\Her_{M_2(C)}(e\oplus k).
\]
Then $H$ is sigma-unital, with no separability assertion required.
If $r\in C$ idealizes $I$, then $r\oplus0$ idealizes $H$.
If, in addition, $L$ is an ideal of $M_2(C)$ and
\[
(rI)\oplus0\subseteq L,\qquad (Ir)\oplus0\subseteq L,
\]
then
\begin{equation}\label{un:support-bound}
(r\oplus0)H+H(r\oplus0)\subseteq H\cap L.
\end{equation}
For $x\in H$, one has
$H\cap\Ideal_{M_2(C)}(x)=\Ideal_H(x)$.
\end{lemma}
\begin{proof}
The defining positive element is strictly positive in its hereditary
algebra, so $H$ is sigma-unital. Since $r e^{1/2}$ and
$e^{1/2}r$ belong to $I$, multiplication of
$(e^{1/2}\oplus k^{1/2})M_2(C)(e^{1/2}\oplus k^{1/2})$
on either side by $r\oplus0$ remains in $H$. These elements are
dense in $H$. Under the additional hypothesis, the same products
also lie in $L$, proving \eqref{un:support-bound}. The last identity
is the ideal correspondence for a hereditary subalgebra: an
approximate unit of $H$ compresses the ambient ideal generated by
$x$ to the ideal it generates in $H$.
\end{proof}
\begin{lemma}[Finite truncation of a proper conjugacy]
\label{un:finite-truncation}
Let $H$ be sigma-unital and let $p,q:S\to\mathcal M(H)$ be
representations with difference in $H$. Let $\theta:S\to H$ be
a homomorphism, compact-valued on the module $H$, and form the diagonal
representations
\[
A_0=p\oplus\theta\oplus\theta\oplus\cdots,
\qquad B_0=q\oplus\theta\oplus\theta\oplus\cdots.
\]
Identical zero summands may be inserted in these diagonals.
If an identity-plus-compact unitary compares $A_0,B_0$ arbitrarily
well on finite sets, then, for every finite $\mathcal F$ and
$\eps>0$, finitely many copies suffice: for some $N,M\geq0$ there is a
unitary $v_N$ equal to the identity outside the initial summand,
the first $N$ repeat summands, and finitely many zero slots, with
\[
\max_{a\in\mathcal F}
\|v_N(p\oplus\theta^{\oplus N}\oplus0^{\oplus M})(a)v_N^*
-(q\oplus\theta^{\oplus N}\oplus0^{\oplus M})(a)\|<\eps.
\]
All resulting errors are compact. If the representations are
weakly equivariant, their compact errors on $S(U)$ belong to the
compacts over $H(U)$.
\end{lemma}
\begin{proof}
Choose $u=1+k$ with error less than $\eps/2$, and put
$R=\max(1,\max_{a\in\mathcal F}\|a\|)$. If $E_N$ denotes the
specified finite-summand projection, choose
\[
\|k-E_NkE_N\|<\eta,
\qquad 0<\eta<\min\{1/4,\eps/(8R)\}.
\]
The operator $z_N=1+E_NkE_N$ is within $\eta$ of a unitary, so it
is invertible and the spectrum of $|z_N|$ is contained in
$[1-\eta,1+\eta]$. Its polar unitary satisfies
\[
v_N=z_N(z_N^*z_N)^{-1/2},\qquad
\|v_N-z_N\|\leq\eta,
\qquad\|v_N-u\|\leq2\eta.
\]
It is the identity off $E_N$ and differs from the identity by a
finite matrix over $H$. Include successively the identical zero
slots as well, if present, so that $E_N\to1$ strongly. The change
in the conjugacy error is at most $4R\eta$. Since the two
representations are diagonal and agree on the omitted tail, the
remaining estimate is exactly the displayed finite-copy estimate.
For localization, a compact operator which is weakly equivariant
on an input in $S(U)$ has all matrix coefficients in $H(U)$,
hence belongs to $H(U)\otimes\K$. All module unitaries preserve
the submodules $E_HH(U)$, so conjugation preserves this statement.
It applies to the compact differences here.
\end{proof}
\subsection{Removing the locally small difference}
\begin{proposition}[Uniqueness with a spare model]\label{un:spare}
Let $p,q:S\to D$ be equal-data models satisfying
$(p-q)(S(U))\subseteq J(U)$ for every open $U$. Let $\chi$ be a
model for a strictly positive multiple of the same data. Then
$p\oplus\chi$ and $q\oplus\chi$ are coordinate approximately
unitarily equivalent, with the matrix and trace conventions of
\Cref{st:folding}.
\end{proposition}
\begin{proof}
Put $C_1=C^*(p(S),q(S))$, and let $I$ be the ideal of $C_1$
generated by the differences $p(a)-q(a)$. This is a separable
ideal. For each open $U$,
\begin{equation}\label{un:difference-ideal}
p(S(U))I+q(S(U))I+Ip(S(U))+Iq(S(U))\subseteq J(U).
\end{equation}
Indeed, modulo $J(U)$, the two maps agree on $S(U)$. Their common
image is an ideal in the algebra generated by both images, because
multiplication from either side by $p(a)$ or $q(a)$ corresponds to
multiplication inside the source ideal. It annihilates each
difference: for $s\in S(U)$,
\[
p(s)(p(a)-q(a))=p(sa)-q(sa)=0\pmod{J(U)},
\]
and the other side is identical. It therefore annihilates the
ideal generated by the differences, proving
\eqref{un:difference-ideal}.
Use the existing-model version of \Cref{sm:stabilizers}, with
the spare $\chi$, to obtain a locally small homomorphism $\theta$.
Impose the coefficient dominations
\begin{equation}\label{un:theta-domination}
p(S(U_x))I+q(S(U_x))I+Ip(S(U_x))+Iq(S(U_x))
\subseteq\Ideal(\theta(x)),\qquad x\in S_+.
\end{equation}
Apply \Cref{sm:coefficient-domination} to the four bounded
bilinear maps $(a,b)\mapsto p(a)b$, $q(a)b$, $bp(a)$, and $bq(a)$,
with $b\in I$. Equation~\eqref{un:difference-ideal} gives their
local hypotheses, and that corollary supplies the displayed inclusion
for every positive $x$. The first version also ensures
\begin{equation}\label{un:theta-range-fullness}
\theta(S(U_x))\subseteq\Ideal(\theta(x)).
\end{equation}
Moreover $\theta$ is represented by the same coordinate maps as
$\chi$, tensored with rational UHF projections of trace tending
to zero.
Choose strictly positive elements $e\in I$ and $k\in\theta(S)$,
and form
\[
H=\Her_{M_2(D)}(e\oplus k).
\]
The maps $p,q$, placed in the first corner and zero in the second,
idealize $H$, by \Cref{un:hereditary-support}. View them as
representations on the module $H$, and then stabilize that module.
Equip $H$ with the order-preserving ideals
\begin{equation}\label{un:H-action}
H(U)=\Ideal_H(0\oplus\theta(S(U))).
\end{equation}
No intersection property of this assignment is needed. From
\eqref{un:theta-domination}, \Cref{un:hereditary-support}, and its
ideal-intersection identity, values of the first-corner maps on
$S(U_x)$ multiply $H$ into
$\Ideal_H(0\oplus\theta(x))$. Thus $p,q$ are weakly equivariant
for \eqref{un:H-action}. By \eqref{un:theta-range-fullness},
$0\oplus\theta(x)$ generates $H(U_x)$, so this compact-valued map
is $X$-full. The algebra $H$ is sigma-unital; it is not required
to be separable.
The multiplier pair agrees modulo $H$, since its differences
belong to $I\oplus0\subseteq H$. Pad the module $H$ by zero
copies to the standard module; the resulting pair class
\eqref{un:pair-cycle} is zero by \Cref{un:KK-nullity}.
The coarse source action is lower semicontinuous, since the
projection $Y\times X\to X$ is open. For
\Cref{un:stable-uniqueness}, place $0\oplus\theta$ in the first
compact corner of the standard module. Its infinite repeat
is, after reindexing, the diagonal of copies of $\theta$ on $H$
with identical zero slots. Thus \Cref{un:finite-truncation}
applies on the original $H$ summands, yielding finite matrices
over $H$, rather than retaining an additional untruncated compact
direction inside each slot. For any finite $\mathcal F$ and
$\eps>0$ this gives a finite $N$ and an identity-plus-finite-matrix
unitary comparing
\begin{equation}\label{un:finite-spare-comparison}
p\oplus\theta^{\oplus N}
\quad\text{and}\quad
q\oplus\theta^{\oplus N}
\end{equation}
within $\eps$, with zero slots permitted.
These estimates hold for the actual ambient elements, as follows.
If $v=1+c$ is the obtained finite unitary and $A,B$ denote the
two finite diagonal maps, then
\begin{equation}\label{un:ambient-error}
vA(a)v^*-B(a)
=(A(a)-B(a))+cA(a)+A(a)c^*+cA(a)c^*.
\end{equation}
Every term belongs to a finite matrix over $H$, because the maps
idealize $H$ and their difference belongs to it. The multiplier
norm of this element is its intrinsic $H$-norm and hence its
ambient norm in the matrix algebra over $D$. This argument does
not assume that restriction of the entire ambient idealizer to
$\mathcal M(H)$ is faithful. Only its isometric restriction to
$H$ is used.
There is also no lifting obstruction for the finite unitary.
Write it as $1+c$ in the unitization of a finite matrix algebra
over $D$, and choose bounded representatives $c_n$ over $Q$.
The two unitary defects of $1+c_n$ tend to zero. For all
sufficiently large $n$ its polar part
\[
v_n=(1+c_n)\bigl((1+c_n)^*(1+c_n)\bigr)^{-1/2}
\]
is a unitary in the corresponding matrix multiplier algebra and
differs from $1+c_n$ by a sequence tending to zero. Thus a strict
quotient estimate, first obtained with a smaller tolerance, holds
at every sufficiently late coordinate.
Finally fix this $N$ before advancing the coordinates. Since the
UHF trace of the projection defining $\theta_n$ tends to zero,
$N$ disjoint equivalent copies fit inside the positive spare
fraction eventually. Using exactly the same $\chi_n$ in those
copies identifies them with \eqref{un:finite-spare-comparison}.
Move the unused spare to a separate stable corner and leave it
unchanged. Consequently the finite conjugacy need not preserve
an internal complement of its original matrix slots. Apply the
coordinate folding and the multiplier permutations from
\Cref{st:folding}, on all current inputs and all these blocks.
This proves eventual finite-test conjugacy of $p\oplus\chi$ and
$q\oplus\chi$. On increasing finite sets with tolerances tending
to zero, choose each coordinate beyond the corresponding
threshold. The resulting single sequence of multiplier unitaries
proves the proposition.
\end{proof}
\begin{proof}[Proof of \Cref{un:uniqueness}]
Split the tensor-first copy of each map along two equivalent UHF
projections of trace $1/2$. Denote its two halves by $p_0,p_1$
and $q_0,q_1$. Both halves have the original model data multiplied
by $1/2$, and their sums are coordinate approximately unitarily
equivalent to the original maps by \Cref{st:folding}.
First compare $p_0\oplus p_1$ with $q_0\oplus p_1$, retaining
$p_1$ as a common spare. Apply \Cref{un:alignment} in the changing
block to put its difference in every appropriate $J(U)$, carrying
the spare through the same diagonal amplifications. Then
\Cref{un:spare} gives the required conjugacy. The coordinate
conjugacies relating the aligned blocks to the originals may be
undone, with their vanishing errors, so the conclusion applies
to the displayed original half-sums. Next compare
$q_0\oplus p_1$ with $q_0\oplus q_1$, using $q_0$ as the common
spare and interchanging the two matrix positions. The same
argument applies. Composing the two coordinate conjugacies and
the folding adjustments proves the assertion on every finite
test set and tolerance. A diagonal choice on a dense increasing
family gives the stated sequence $u_n$.
Finally, any two c.p.c. lifts of a fixed model differ in norm by a
sequence tending to zero on each input. The conclusion therefore
holds for the lifts specified in the statement.
\end{proof}
==== END The-trace-cone-classifies-Razak-Jacelon-stabilizations//sections/05-uniqueness.tex ====
==== BEGIN The-trace-cone-classifies-Razak-Jacelon-stabilizations//sections/06-tracial-cone.tex ====
\section{Tracial approximation on a cone}\label{tc:section}
Throughout this section, $Y=(0,1]$, $m$ is a finite Borel measure of full
support on $Y$, and $S=C_0(Y)\otimes P$. We retain the invariant maps
$F:T(Q)\to T(P)$ and $G:\Cu(P)\to\Cu(Q)$, so that
\begin{equation}\label{tc:transport}
d_\tau(G[a])=d_{F(\tau)}(a).
\end{equation}
We seek a completely positive map that is multiplicative modulo the
local error ideals. It must carry two kinds of information. Its square
moments will identify the prescribed trace wherever that trace is densely
finite. The ideal generated by its image of a nonzero elementary tensor
must also contain the corresponding constant ideal. This second condition will prevent the
corrected map from turning an infinite part of a weight into a finite one.
Matrix traces are unnormalized. The rational averages used below have a
separate, normalized UHF tensor factor; \Cref{st:folding} identifies their
ranges with $Q$ with precisely this trace convention.
Choose a countable family $\mathcal E$ of positive elementary tensors
$s=f\otimes a$ as follows. In each ideal in the countable basis, and in
each finite union of those ideals, take the positive rational cuts of a
norm-dense set of positives. Retain slightly larger positive pre-cuts in the
same ideal. Do the same for positive functions in $C_0(Y)$, and include
the powers and further cuts of the factors that occur below. Thus the
local linear spans of $\mathcal E$ are dense, and its rectangle supports
exhaust the open subsets of $Y\times X$. All elementary tests used in
this section may be included in this one countable family.
For a test $s_i=f_i\otimes a_i$ supported over an open set $U_i$, choose,
once and for all, a positive controller $b_i\in Q(U_i)$ of compact Cuntz
support such that
\begin{equation}\label{tc:controller}
G[a_i]\leq[b_i].
\end{equation}
The larger pre-cuts allow this choice inside $Q(U_i)$, with room for a
further compactly supported upper controller there. When needed, the
same test is listed with different containing basis opens. This ensures
that the controllers required for a local assertion lie in that open,
as in \Cref{seq:diagonal}.
\begin{theorem}\label{tc:rough-model}
There is a completely positive contraction $h:S\to D=Q_\infty$ with the
following properties.
For every open $U\subseteq X$, $s\in S(U)$, and $z\in S$,
\begin{equation}\label{tc:mixed}
h(sz)-h(s)h(z),\quad h(zs)-h(z)h(s)\ \in J(U).
\end{equation}
If $s=f\otimes a\in\mathcal E$ is controlled inside $U$, and $\rho$ is a
regularized limit trace whose restriction $\sigma=\rho|_Q$ is densely
finite on $Q(U)$, then
\begin{equation}\label{tc:square}
\rho(h(s)^2)=\left(\int_Y f^2\,dm\right)F(\sigma)(a^2).
\end{equation}
Finally, for every $f\otimes a\in\mathcal E$ with $f\ne0$,
\begin{equation}\label{tc:lower-ideal}
Q(U_a)\subseteq\Ideal_D\bigl(h(f\otimes a)\bigr).
\end{equation}
The test family may be enlarged in advance by any countable family of
the controlled elementary tests just described.
\end{theorem}
For a finite-stage completely positive contraction $H:S\to M_k(Q)$,
we keep three trace coordinates for each positive input $s$: the trace
values of $H(s)$, $H(s^2)$, and $H(s)^2$. Matching the last
two makes the positive Schwarz defect small in trace. The first bounds
the square mass that could otherwise disappear when a coordinate trace
is regularized.
Trace approximation alone does not give the lower ideal inclusion. For
that we also approximate positive expressions
$yv^*g(H(s))vy$ by $y^2$, where $y$ is a positive constant from the prescribed
ideal of $Q$, $v\in M_{k,1}(Q)$ is a contraction, and the continuous function
$g:[0,\infty)\to[0,1]$ vanishes below a fixed positive cutoff. Their norm approximation will force a lower rank bound even at
weights for which the controller has infinite rank. We first realize
these finite data in von Neumann algebras, then approximate them in $Q$
and average. A final diagonal choice retains the fixed controllers and
cutoffs.
\subsection{Realization in von Neumann algebras}
We first record explicitly the trace-preserving embedding needed for
extended weights. The embedding is not required to be unital.
We write $R$ for the hyperfinite factor of type $\mathrm{II}_1$ and
$R_\infty=R\,\overline\otimes\,B(\ell^2)$ for the hyperfinite factor
of type $\mathrm{II}_\infty$. Their uniqueness is part of Connes's
classification \cite{Connes}; see also the distinct contemporary
report \cite[Theorems~4.2.1 and~4.3.1]{ConnesReport}.
\begin{lemma}[Trace-preserving semifinite embedding]\label{tc:semifinite-embedding}
Let $N$ be a hyperfinite von Neumann algebra with separable predual and a
faithful normal semifinite trace $T$. There is a normal injective
homomorphism from $N$ into the hyperfinite factor $R_\infty$ of type
$\mathrm{II}_\infty$ preserving $T$. Consequently, such an embedding
exists into any prescribed infinite corner of a separably acting factor
of type $\mathrm{II}_\infty$.
\end{lemma}
\begin{proof}
The central decomposition of a semifinite hyperfinite algebra gives a
countable direct sum of algebras
\[
L^\infty(X_j,\mu_j^0)\,\overline\otimes\,N_j,
\qquad
N_j\in\{M_n,\ B(\ell^2),\ R,\ R_\infty\}.
\]
Here the measure spaces are standard and the factor traces are the
usual matrix trace or the standard finite or semifinite trace. This is
the semifinite form of the hyperfinite central decomposition; see
\cite[Section~6, p.~49, after (6.48) and (6.50)]{HRS}. Relative to these
traces and a finite measure $\mu_j^0$ in the measure class, $T$ has a
central Radon--Nikodym density $h_j$. Faithfulness and semifiniteness
give $0<h_j<\infty$ almost everywhere. Replacing $\mu_j^0$ by
$\mu_j=h_j\mu_j^0$ absorbs the density and leaves a standard
$\sigma$-finite measure space.
There is a normal, trace-preserving injection
\[
L^\infty(X_j,\mu_j)\longrightarrow
L^\infty([0,\infty),dt).
\]
Indeed, partition $X_j$ into countably many pieces of finite measure.
On each piece assign an interval of the appropriate length to each
atom; the remaining diffuse part is isomorphic, modulo null sets, to
the remaining interval. Pullback by the resulting measure-preserving
map, extended by zero outside the union of the intervals, gives the
injection. Moreover,
$L^\infty([0,\infty))$ embeds trace-preservingly as
$L^\infty([0,1])\overline\otimes\ell^\infty$ in
$R\overline\otimes B(\ell^2)=R_\infty$.
Each standard factor $N_j$ embeds trace-preservingly in $R_\infty$:
use $1_R\otimes M_n$, $1_R\otimes B(\ell^2)$,
$R\otimes e_{11}$, or the identity map, respectively. Tensor these
embeddings with the preceding abelian embeddings and use the
trace-preserving isomorphism
$R_\infty\overline\otimes R_\infty\cong R_\infty$.
The countably many summands can be placed in the diagonal corners of
$R_\infty\overline\otimes B(\ell^2)$. The orthogonal strong sum is
normal, injective, and trace preserving.
Finally, in an infinite corner of an arbitrary factor of type
$\mathrm{II}_\infty$, take a projection of trace one, nested unital
matrix algebras $M_{2^n}$ in its finite corner, and their weak closure.
They give a trace-preserving copy of $R$. Matrix amplification in the
infinite corner gives $R_\infty$. Composition completes the proof.
\end{proof}
\begin{lemma}[Realization in a factor]\label{tc:factor-realization}
Let $\pi:Q\to M$ be a nondegenerate representation with weakly dense
range in a separably acting factor. There is a homomorphism
$l:S\to M$ with the following rank data. If $M$ is semifinite with
faithful dimension trace $T$, then
\begin{equation}\label{tc:factor-rank}
d_T(l(f\otimes a))
=m(\{f>0\})d_{F(T\circ\pi)}(a)
\qquad(f,a\geq0).
\end{equation}
In a type III factor the corresponding support projection is nonzero
exactly when the transported ideal data prescribe a nonzero support.
\end{lemma}
\begin{proof}
Stability of $Q$ supplies multiplier isometries whose normal images
show that $M$ is properly infinite. Since $M$ is separably acting,
the usual scalar trace-dimension comparison applies in its semifinite
factor cases; in its type~III case all nonzero projections are
equivalent \cite[III.1.7.9--10 and III.1.3.7]{Blackadar}. Suppose first that $M$ is of type
I. The rank function $d_{T\circ\pi}$ on $\Cu(Q)$ takes only the values
$0$ and $\infty$. Otherwise real scalar divisibility of $\Cu(Q)$,
applied to a class of finite nonzero integer rank, would give a class
of noninteger rank in a type I factor. Thus the right side of
\eqref{tc:factor-rank} also takes only these two values. Its zero ideal
on $P$ is the ideal transported from $\ker\pi$. A faithful separable
representation of the corresponding quotient of $S$, repeated with
infinite multiplicity and embedded in $M$, realizes the required ranks.
For type III the same construction realizes the zero ideal; all
nonzero support projections are equivalent. The assertion is also
immediate when the prescribed quotient is zero.
Suppose now that $M$ has type $\mathrm{II}_\infty$, and put
\[
\nu=m\otimes F(T\circ\pi),\qquad I=\Fin(\nu),
\qquad K=\Ker(\nu).
\]
Here $\Fin(\nu)$ is the closed ideal on which $\nu$ is densely finite,
and $K\subseteq I$ is its zero ideal. The trace representation of
$I/K$ has weak closure $N$ carrying the faithful normal semifinite
trace induced by $\nu$. Nuclearity of $I/K$ makes $N$ hyperfinite
\cite{Connes}; the nuclear-representation consequence is also stated
in \cite[Theorem~4.1 and the following paragraph]{ConnesReport}.
The algebra $N$ has separable predual. Its nondegenerate
representation extends from $I$ to $S$ by the action of $S$ on the
ideal $I$. \Cref{tc:semifinite-embedding} embeds this representation
trace-preservingly in one infinite corner of $M$.
In an orthogonal infinite corner, choose a copy of $B(\ell^2)$ whose
minimal projections have a fixed positive finite trace. In this copy
place an infinite multiple of a faithful separable representation of
$S/I$. On $I$ this second summand is zero. If $x\in S_+$ does not
belong to $I$, its second image has infinite trace and infinite support
rank. This is exactly what is required, since $\nu(x)=\infty$ off
$I$. On $I$, the first summand gives $\nu$ and all its ranks, including
infinite values. The combined homomorphism therefore satisfies
$T\circ l=\nu$, has zero ideal $K$, and proves
\eqref{tc:factor-rank}. If $I=0$ or $I=S$, simply omit the empty
summand.
\end{proof}
Factorwise existence is enough only if the maps can be assembled
measurably. We encode their rank requirements by partial isometries
between support projections. These witnesses assemble along with the
maps, and later allow each selected normal trace to evaluate the required
ranks directly.
\begin{lemma}[Measurable realization]\label{tc:measurable-realization}
Let $\pi$ be a nondegenerate separably acting representation of $Q$,
and put $M=\pi(Q)''$. There is a homomorphism $l:S\to M$ whose
factorwise ranks are those in \Cref{tc:factor-realization}, on any
prescribed countable elementary cut-test family. In particular, it
may be chosen on the entire family $\mathcal E$ and its required cuts.
\end{lemma}
\begin{proof}
Use the standard central decomposition of $M$, modulo null sets, into
separably acting factors $M_x$ \cite[III.5.1.14--15]{Blackadar}. We explain the measurable selection
in a form that does not choose a normalization of the factor traces.
For each elementary test $s_j=f_j\otimes a_j$ fix $c_j\in Q_+$ with
\begin{equation}\label{tc:transported-representative}
[c_j]=m(\{f_j>0\})G[a_j].
\end{equation}
Realification and stability provide these representatives. Code a
contractive homomorphism on a countable dense rational $*$-algebra in
$S$ by its values in the strong-$*$ bounded balls of the factor field.
Include norm-convergent approximations to the chosen positive tests.
Multiplication on bounded balls, adjoints, norm bounds, and membership
in the measurable algebra field give countably many Borel conditions.
Membership may be checked against the strong-$*$ dense measurable
sections of the field. Adjoin contraction coordinates $w_j\in M_x$
subject to
\begin{equation}\label{tc:selection-witness}
w_j^*w_j=\supp(l_x(s_j)),\qquad
w_jw_j^*=\supp(\pi_x(c_j)).
\end{equation}
The support map is the strong limit of
$t\mapsto t(t+1/n)^{-1}$, so these are again Borel conditions.
Thus the augmented admissible-code relation is Borel in a countable
product of standard measurable bounded operator fields. Its sections
are nonempty by \Cref{tc:factor-realization}: equal factor dimensions
give the partial isometries in \eqref{tc:selection-witness}. Apply
Jankov--von Neumann measurable uniformization to this augmented analytic
relation, retaining both the homomorphism coordinates and
all the contraction witnesses $w_j$. The selector is measurable for
the completed central measure. Replace its countably many coordinates
by measurable representatives and discard one common null set on which
any selected algebraic equation or support identity fails.
This is precisely the measured-field setting, as distinguished from
an everywhere Borel field in \cite[Section~8, especially Lemma~8.6(1),
and the proof of Proposition~3.2]{VaesWouters}.
The uniform operator bounds make the selected values, including the
$w_j$, decomposable operators. Their countably many algebraic relations
define a homomorphism $l:S\to M$, and the assembled witnesses establish
\eqref{tc:selection-witness} in $M$. Every elementary cut test needed
below is included in the code, so its prescribed rank data hold in
this field.
\end{proof}
Here is the finite-trace consequence of this construction that we will
use. It specifies the normal trace associated to each selected weight
and the finite constants through which the rank data are evaluated.
\begin{lemma}[Realization for finitely many weights]\label{tc:finite-trace-realization}
Fix finitely many controlled tests $s_i=f_i\otimes a_i$ and finitely
many traces $\tau$ for each $i$, each satisfying $d_\tau(b_i)<\infty$.
Take a separably acting representation $\pi$ containing the induced
trace representations of their densely finite ideals, as well as any
prescribed finite family of representations for bounded functionals
on $Q$. The homomorphism $l$ in \Cref{tc:measurable-realization} may
be chosen so that, for the induced normal trace $\widetilde\tau$ on
$M=\pi(Q)''$,
\begin{equation}\label{tc:finite-values}
\widetilde\tau(l(s_i^k))
=\left(\int f_i^k\,dm\right)F(\tau)(a_i^k),\qquad k=1,2.
\end{equation}
This assertion requires no commutation of $F$ with a direct integral
of extended weights.
\end{lemma}
\begin{proof}
Fix one selected $\tau$ and put $I=\Fin(\tau)$. Its trace representation
$\pi_\tau$ on $I/\Ker(\tau)$ has weak closure $N_\tau$ with faithful
normal semifinite trace $T_\tau$. The nondegenerate extension of
$\pi_\tau$ to $Q$ still has image in $N_\tau$: for an approximate unit
$(u_\lambda)$ of $I$, the operators
$\pi_\tau(q)\pi_\tau(u_\lambda)\in\pi_\tau(I)$ converge strongly to
$\pi_\tau(q)$. Hence $\pi_\tau(Q)''=N_\tau$.
The projection onto this representation summand commutes with
$M=\pi(Q)''$. Restriction to that summand is therefore a normal
surjective homomorphism $\theta_\tau:M\to N_\tau$. Its kernel has
the form $(1-z_\tau)M$ for a central projection $z_\tau$, and its
restriction to $z_\tau M$ is a normal isomorphism. Define
\[
\widetilde\tau=T_\tau\circ\theta_\tau.
\]
This is a normal semifinite trace, zero on $(1-z_\tau)M$, and it agrees
with $\tau$ on $\pi(I)_+$. Normality also gives
$d_{\widetilde\tau}(\pi(c))=d_\tau(c)$ for $c\in I_+$.
The projections $z_\tau$ for different selected weights may overlap:
each trace is defined through its own normal quotient, with no
disjointness or additional contribution from other representation
summands. The construction includes the zero trace representation.
Agreement with $\tau$ outside $I$ is neither asserted nor needed.
Include the rectangle tests
$(f_i-r)_+\otimes(a_i-t)_+$ for rational $r,t\geq0$, and choose
$c_{r,t}\in Q_+$ with
\[
[c_{r,t}]=m(\{f_i>r\})G[(a_i-t)_+].
\]
Their ranks are at most $m(Y)d_\tau(b_i)<\infty$, so $c_{r,t}\in I$.
Retaining the selected witness coordinates in
\eqref{tc:selection-witness} gives, in $M$, equivalence of the support
of $l((f_i-r)_+\otimes(a_i-t)_+)$ with $\supp(\pi(c_{r,t}))$.
Applying $\widetilde\tau$ directly to these equivalent projections gives
\[
\widetilde\tau\bigl(\supp l((f_i-r)_+\otimes(a_i-t)_+)\bigr)
=m(\{f_i>r\})d_{F(\tau)}((a_i-t)_+).
\]
In particular, $\supp(l(s_i))$ has finite $\widetilde\tau$-trace.
In the support corner of $l$, the multiplier images of $f_i\otimes1$
and $1\otimes a_i$ commute, and their product is $l(s_i)$. The last
display specifies their finite joint spectral measure on every upper
rectangle in the positive quadrant. It is the product of the
spectral measure of $f_i$ under $m$ and that of $a_i$ under $F(\tau)$.
Increasing positive step approximations to the product function
$(u,v)\mapsto u^kv^k$, using rational spectral partitions away from
zero, therefore give \eqref{tc:finite-values} by normal monotone
convergence, for $k=1,2$. All rank identities used here were evaluated
on constants in $\Fin(\tau)$.
\end{proof}
\subsection{Simultaneous finite approximation}
The von Neumann algebra realizations attain the prescribed trace and
support data. We now approximate finitely many of those data inside
$Q$. The approximations must retain a finite rank bound over an entire
trace cap, since pointwise trace approximation would not control the
local error ideals after diagonalization.
Fix finitely many tests $s_i=f_i\otimes a_i$, indexed by $1\leq i\leq n$,
with their fixed controllers. Set
\begin{equation}\label{tc:caps-targets}
K_i=\{\tau\in T(Q):d_\tau(b_i)\leq1\},\qquad
t_i^{(k)}(\tau)=\left(\int f_i^k\,dm\right)F(\tau)(a_i^k).
\end{equation}
By \Cref{un:trace-cap}, $K_i$ is compact and convex. The functions
$t_i^{(k)}$ are finite continuous affine functions on $K_i$.
We impose finitely many lower comparisons, whose indices $j$ refer to
some of the same tests. Choose $\delta_j>0$ and $r_j>0$ with
\begin{equation}\label{tc:lower-data}
r_j<m(\{f_j>\delta_j\}),\qquad
[y_j]=r_jG[(a_j-\delta_j)_+],\qquad y_j\in Q_+.
\end{equation}
Rescale representatives so that $\|y_j\|\leq1$. Choose a continuous
function $g_j:[0,\infty)\to[0,1]$ and constants
\begin{equation}\label{tc:fixed-slack}
0<\eta_j<\theta_j<\delta_j^2,
\qquad g_j|_{[0,\eta_j]}=0,
\qquad g_j|_{[\theta_j,\infty)}=1.
\end{equation}
These constants are fixed before any finite approximation or averaging.
A candidate consists of a completely positive contraction
$H:S\to M_k(Q)$ and contractions $v_j\in M_{k,1}(Q)$, with the following
whole-cap support condition: for each $i$ there is a finite constant
$C_{H,i}$ such that, for every $\tau\in T(Q)$,
\begin{equation}\label{tc:candidate-rank}
\begin{split}
d_{\tau\otimes\Tr}(H(s_i))&\leq C_{H,i}d_\tau(b_i),\\
d_{\tau\otimes\Tr}(H(s_i^2))&\leq C_{H,i}d_\tau(b_i).
\end{split}
\end{equation}
The constants may depend on the candidate. Its vector in the real
locally convex space
\[
\prod_{i=1}^n C(K_i,\R)^3\ \times\ \prod_j Q_{\mathrm{sa}}
\]
is
\begin{equation}\label{tc:candidate-vector}
\left(
(\tau(H(s_i)),\tau(H(s_i^2)),\tau(H(s_i)^2))_i;
(y_jv_j^*g_j(H(s_j))v_jy_j)_j
\right).
\end{equation}
Here and below $\tau$ on matrices means $\tau\otimes\Tr$.
The support bounds and \Cref{un:trace-cap} make all three trace
coordinates continuous. No bound uniform over all candidates is
needed for this assertion.
\begin{proposition}[Simultaneous finite approximation]\label{tc:finite-convexity}
The target vector
\begin{equation}\label{tc:target-vector}
\bigl((t_i^{(1)},t_i^{(2)},t_i^{(2)})_i;(y_j^2)_j\bigr)
\end{equation}
belongs to the closed convex hull of the candidate vectors.
\end{proposition}
\begin{proof}
\smallskip\noindent\emph{Reduce a separator to finitely many weights.}
Suppose a continuous real linear functional strictly separates the
target from this convex hull. On each $C(K_i,\R)$ coordinate it is
integration against a finite signed measure. Split each measure into
its positive and negative parts. By \Cref{un:barycenter}, each nonzero
part has a barycenter in $K_i$ reproducing every finite continuous
affine evaluation used here; its mass is an ordinary scalar
coefficient. This holds simultaneously for the candidate evaluations
and the target evaluations, including candidates with different
support constants. Thus the separating functional involves only
finitely many selected traces. On the algebra coordinates it involves
finitely many bounded selfadjoint functionals.
\smallskip\noindent\emph{Realize the selected values exactly.}
Choose a faithful separably acting representation $\pi$ of $Q$
containing representations for these bounded functionals and the
induced representations of the densely finite ideals of the selected
traces. Write $M=\pi(Q)''$ and identify $Q$ with its image.
\Cref{tc:measurable-realization,tc:finite-trace-realization} give a
homomorphism $l:S\to M$ attaining the selected trace values. We next
attain the lower norm coordinates in $M$.
Let $q_j$ be the support projection of
$l((f_j-\delta_j)_+\otimes(a_j-\delta_j)_+)$. Functional calculus in
the two commuting factors gives
\[
q_j\leq1_{(\delta_j^2,\infty)}(l(s_j)).
\]
The rank data and \eqref{tc:lower-data} give, in every factor,
$\supp(y_j)\precsim q_j$. This comparison also covers zero and
infinite dimensions and the type III parts. Comparison of projections
in a direct integral therefore gives a partial isometry $v_j\in M$
with initial projection $\supp(y_j)$ and final projection at most
$q_j$. Hence
\begin{equation}\label{tc:exact-norm-signal}
y_jv_j^*g_j(l(s_j))v_jy_j=y_j^2.
\end{equation}
\smallskip\noindent\emph{Approximate inside fixed supports.}
It remains to approximate these finite data by candidates satisfying
\eqref{tc:candidate-rank} on the entire caps. A tested value must vanish
exactly in a block where its controller vanishes; strong approximation
alone would not retain this condition. We therefore split into central
sectors and factor each sector map through the required source quotient
before approximating it. Let $z_i$ be the central
support of $b_i$ in $M$, and form the finitely many central sectors
\[
z_E=\prod_{i\in E}z_i\prod_{i\notin E}(1-z_i),
\qquad E\subseteq\{1,\ldots,n\}.
\]
If $i\notin E$, the rank data and $G[a_i]\leq[b_i]$ imply
$z_El(s_i)=0$. Let $J_E^S$ be the ideal of $S$ generated by these
$s_i$, and let
\[
I_E=\bigcap_{i\in E}\Ideal_Q(b_i),
\qquad I_{\varnothing}=Q.
\]
The weak closure of $I_E$ has central support dominating $z_E$.
Choose positive contractions $e_E\in I_E$ with compact Cuntz support,
tending strongly to the identity on this sector. The maps
\begin{equation}\label{tc:sector-compression}
\psi_E(s)=e_Ez_El(s)e_E
\end{equation}
take values in $\Her(e_E)''$, are completely positive contractions,
and factor through $S/J_E^S$. This quotient is taken before the
approximations below. The element $e_E$ itself need not vanish outside
the sector; \eqref{tc:sector-compression} does vanish there.
Write
\[
D_0(s)=\operatorname{diag}_E(z_El(s)),\qquad
D_e(s)=\operatorname{diag}_E(e_Ez_El(s)e_E).
\]
Thus $D_0$ is the exact sector representation and $D_e$ is its completely
positive compression. As the $e_E$ increase along approximate units,
$D_e(s)\to D_0(s)$ strongly-$*$ for each $s$.
For the selected trace belonging to test $i$, the sector contributes
zero if $i\notin E$. If $i\in E$, the positive operators
$z_El(s_i)$ and $z_El(s_i^2)$ are integrable, and $z_El(s_i)$ is square
integrable, by \eqref{tc:finite-values}. Strongly converging bounded
approximate units therefore make the compressions converge in the
required $L^1$ and $L^2$ seminorms. For the $L^1$ assertion apply
$L^2$ convergence to the square root of the integrable positive
operator and use Cauchy--Schwarz; for the square evaluation apply it
directly to $z_El(s_i)$.
For the lower signal set $V_j=(z_Ev_j)_E$. This column is a
contraction, and, because $g_j(0)=0$, the exact sector representation
satisfies
\[
y_jV_j^*g_j(D_0(s_j))V_jy_j=y_j^2.
\]
Bounded strong-$*$ functional calculus gives
$g_j(D_e(s_j))\to g_j(D_0(s_j))$. Fix the $e_E$ sufficiently far out
that replacing $D_0$ by $D_e$ makes the errors small against the selected
normal functionals, as well as meeting the preceding finite trace
requirements. The compressed expression need not equal $y_j^2$ exactly.
We give the remaining approximation explicitly. Nuclearity of the
quotient $R_0=S/J_E^S$ provides completely positive contractive
factorizations through finite matrix algebras approximating its
identity on the finite tests \cite{ChoiEffros}; see also
\cite[Corollary~3.3]{HanPaulsen}. For nonunital $R_0$, apply the latter
unital statement to its nuclear unitization. Restrict the incoming
map to $R_0$ and compress the outgoing map $\beta$ to
$x\mapsto e\beta(x)e$, where $e\in R_0$ is a positive contractive
approximate-unit element. Choose $e$ first so that
$\|eae-a\|<\delta/2$ on the finite tests, and then choose the
factorization with error less than $\delta/2$ there. Contractivity
bounds the total error by $\delta$, and the compressed map takes
values in $R_0$. Compose this outgoing map with $\psi_E$.
A completely positive map from $M_d$ into $\Her(e_E)''$ is specified
by its positive Choi matrix. Approximate the square root of that
matrix strongly-$*$, with bounded norm, by matrices over
$\Her(e_E)$; the products give positive Choi matrices over the
hereditary algebra. If the resulting map $\beta$ has
$t=\beta(1)$, replace it by
\[
x\longmapsto \max(1,t)^{-1/2}\beta(x)\max(1,t)^{-1/2}.
\]
The functional calculus is in the unitization. This replacement is
completely positive and contractive, stays in $\Her(e_E)$, and
retains the strong-$*$ limit, because the limiting unit value is at
most one. We have obtained completely positive contractions
\[
H_E:S/J_E^S\longrightarrow\Her(e_E)\subseteq Q
\]
approximating \eqref{tc:sector-compression} on the required tests.
The support element $e_E$ was fixed before this Choi-matrix
approximation, and every approximating value remains in $\Her(e_E)$.
For each $i\in E$, compact Cuntz support inside $I_E$ gives a finite
integer $N_{E,i}$ with $[e_E]\leq N_{E,i}[b_i]$. Thus the support of
every output in this block has rank at most
$N_{E,i}d_\tau(b_i)$, for every extended trace $\tau$. On the
selected traces this is one fixed finite normal trace support.
Bounded strong-$*$ convergence on that support gives convergence of
the trace and square-trace evaluations, with no loss of mass. For
$i\notin E$, the quotient-first factorization kills $s_i$ and $s_i^2$
exactly, so no trace-convergence argument is required in that block.
This proves the trace approximation, as well as the required
uniformity on the whole caps.
Put the sector maps in separate matrix blocks:
$H=\bigoplus_E H_E$. This is a completely positive contraction since
the norm of a direct sum is the maximum of the norms. Its
unnormalized matrix trace is the sum of the sector traces. The blocks
with $i\notin E$ kill $s_i$ and $s_i^2$ exactly, while
\[
d_\tau(H(s_i)),\ d_\tau(H(s_i^2))
\leq\left(\sum_{E\ni i}N_{E,i}\right)d_\tau(b_i).
\]
The maps $H_E$ approximate the fixed compressed map $D_e$,
not the exact map $D_0$. Rectangular Kaplansky density approximates the
columns $V_j$ strongly-$*$ by contraction columns over $Q$. Bounded
strong-$*$ functional calculus and multiplication therefore make the
resulting algebra coordinates arbitrarily close, against the selected
normal functionals, to those for $D_e$. Combined with the preceding
choice of the $e_E$, this approximates $y_j^2$ to the required functional
accuracy. Those functionals are normal in $\pi$ by construction.
Thus candidate vectors approximate the value of every finite
functional obtained from a supposed separator, arbitrarily closely
to the target value. This contradicts strict separation and proves
the proposition.
\end{proof}
\subsection{Averaging and the lower rank bound}
The preceding proposition gives finite convex combinations with
arbitrarily small uniform errors on the caps and arbitrarily small
norm errors in the algebra coordinates. The convex coefficients can
be made rational: only finitely many bounded vectors are involved in
any one combination. If their common denominator is $N$, put the
candidates, with their rational multiplicities, in the $N$ diagonal
blocks of a normalized matrix subalgebra of $\UHF$. First zero-pad
their internal matrix spaces to a common size. Denote the resulting
completely positive contraction by $H_{\mathrm{av}}$. Then, for
$k=1,2$,
\begin{equation}\label{tc:average-traces}
\begin{split}
\tau(H_{\mathrm{av}}(s_i^k))
&=\sum_\alpha\lambda_\alpha\tau(H_\alpha(s_i^k)),\\
\tau(H_{\mathrm{av}}(s_i)^2)
&=\sum_\alpha\lambda_\alpha\tau(H_\alpha(s_i)^2).
\end{split}
\end{equation}
Only the $\alpha$ direction is normalized. Internal sector and
stabilizing matrix directions keep the unnormalized trace. Folding
as in \Cref{st:folding} therefore gives the same formulas for a map
with range $Q$.
\begin{lemma}[A lower rank bound preserved by averaging]\label{tc:rank-average}
Fix lower data $y,g,\eta$ as in
\eqref{tc:lower-data}--\eqref{tc:fixed-slack}. If a finite rational
average satisfies
\[
\left\|\sum_\alpha\lambda_\alpha
yv_\alpha^*g(H_\alpha(s))v_\alpha y-y^2\right\|
<\eps^2/2,
\]
then for every extended trace on $Q$,
\begin{equation}\label{tc:rank-bound}
d_\tau((y-\eps)_+)
\leq \frac{2\|y\|^2}{\eps^2}\,
d_\tau((H_{\mathrm{av}}(s)-\eta)_+).
\end{equation}
The constant and the cut $\eta$ are independent of the number of
candidates, their multiplicities, and their internal matrix sizes.
\end{lemma}
\begin{proof}
Write the displayed average as $T$ and let
$p=1_{(\eps,\infty)}(y)$ in the bidual. The norm estimate gives
$pTp\geq(\eps^2/2)p$. Since the $v_\alpha$ are contractions and
$0\leq g(t)\leq1_{(\eta,\infty)}(t)$, positivity and traciality give
\[
\frac{\eps^2}{2}\tau(p)
\leq \|y\|^2\sum_\alpha\lambda_\alpha
d_\tau((H_\alpha(s)-\eta)_+).
\]
The sum is exactly the rank on the right of
\eqref{tc:rank-bound}, by the normalized averaging convention.
Here is a justification when $\tau$ is not densely finite. If the
right side is infinite, there is nothing to prove. If it is finite,
discard zero coefficients. Every support of $g(H_\alpha(s))$ then
has finite rank. The norm estimate gives
$(y^2-\eps^2/2)_+\precsim T$, and $T$ is Cuntz below the finite
direct sum of the $g(H_\alpha(s))$, by its finite row factorization.
Since $p$ is the support of $(y^2-\eps^2)_+$, it has finite rank
and belongs to the normal trace's finite part. One may initially
use a slightly smaller spectral projection inside $p$.
The displayed computation is now a computation for the induced
normal trace on this finite part. Let the additional cut slack decrease to zero. This gives
the inequality for $p$ by normal monotone convergence and proves
the assertion for every extended weight.
\end{proof}
\subsection{The diagonal choice}
\begin{proof}[Proof of \Cref{tc:rough-model}]
\smallskip\noindent\emph{Fix the comparisons before forming averages.}
Fix the test family, its larger pre-cuts, its local controllers, and
all lower data before choosing the sequence. For each nonzero
$f\otimes a\in\mathcal E$, take countably many pairs $(\delta,r)$
with
\[
\delta\downarrow0,\qquad 0<r<m(\{f>\delta\}),
\]
and positives $y$ representing $rG[(a-\delta)_+]$. Include all
positive rational cuts of these $y$ in the lower requirements. Their
ideals exhaust $Q(U_a)$: multiplication by a positive real scalar
does not change a Cuntz ideal, the cuts of $a$ exhaust its ideal, and
$G$ identifies ideals. For each requirement fix its function $g$,
its lower cutoff $\eta$, and its source cutoff $\eps$. In particular,
none of these cutoffs depends on the eventual averaging denominator.
At stage $n$, apply \Cref{tc:finite-convexity} to the first $n$ trace
tests and first $n$ lower requirements. Choose rational averages and
fold them into $Q$, obtaining a completely positive contraction
$h_n:S\to Q$. Given any prescribed sequence $\gamma_n\downarrow0$,
the trace errors can be made to satisfy, for $i\leq n$,
\begin{equation}\label{tc:coordinate-errors}
\begin{split}
|\tau(h_n(s_i))-t_i^{(1)}(\tau)|
&\leq\gamma_n d_\tau(b_i),\\
|\tau(h_n(s_i^2))-t_i^{(2)}(\tau)|
&\leq\gamma_n d_\tau(b_i),\\
|\tau(h_n(s_i)^2)-t_i^{(2)}(\tau)|
&\leq\gamma_n d_\tau(b_i)
\end{split}
\end{equation}
whenever $d_\tau(b_i)<\infty$. Indeed, the uniform estimate on $K_i$
rescales when this rank is nonzero; when it is zero, apply the cap
estimate to every positive scalar multiple of $\tau$ and let the
scalar increase. All evaluations in that argument are finite, by
the support bounds. Simultaneously choose the algebra norm errors
below $\eps^2/2$ for each of the first $n$ fixed lower requirements.
\Cref{tc:rank-average} then supplies the corresponding lower rank
inequalities with their fixed constants.
\smallskip\noindent\emph{Localize the multiplicative defects.}
The coordinate maps define a completely positive contraction
$h:S\to D$. Put
$\Delta_{n,i}=h_n(s_i^2)-h_n(s_i)^2\geq0$ by Schwarz's inequality.
For $t>0$, \eqref{tc:coordinate-errors} gives
\begin{equation}\label{tc:defect-rank}
d_\tau((\Delta_{n,i}-t)_+)
\leq\frac{2\gamma_n}{t}d_\tau(b_i).
\end{equation}
For zero controller rank this follows by the same rescaling argument;
for infinite controller rank the upper bound is automatic. Thus
$h(s_i^2)-h(s_i)^2\in J(U_i)$. In $D/J(U_i)$ the positive element
$s_i$ belongs to the multiplicative domain of the induced completely
positive contraction. Hence both mixed defects with $s_i$ belong
to $J(U_i)$, for every $z\in S$.
For an arbitrary open $U$, use the tests localized in basis ideals
inside $U$. Their linear spans are dense in $S(U)$. Inclusion and
closedness of the ideals $J(V)\subseteq J(U)$ therefore give
\eqref{tc:mixed} for all $s\in S(U)$. The local scheduling in
\Cref{seq:diagonal} ensures the same conclusion when a controller is
first selected in a smaller open: dominate it by a finite multiple
of a fixed controller in the chosen larger open, and reduce the
stage error before passing to the next row. These multiplicities
never change an already fixed cut threshold or lower comparison.
\smallskip\noindent\emph{Recover the square moments.}
Let $\rho$ now be a regularized limit of coordinate weights $\tau_n$,
with $\sigma=\rho|_Q$ densely finite on the chosen controlling ideal
$Q(U_i)$. Compact Cuntz support, and the larger controller fixed at
the start, imply that $d_{\tau_n}(b_i)$ is bounded along the defining
ultrafilter. The constant restrictions converge to $\sigma$ in the
trace topology of \Cref{seq:regularized}. Bounded-rank continuity
from \Cref{un:trace-cap}, applied also after $F$ using
\eqref{tc:controller}, gives
\[
\lim_\omega t_i^{(k)}(\tau_n)=t_i^{(k)}(\sigma),\qquad k=1,2.
\]
Consequently \eqref{tc:coordinate-errors} gives convergence of the
uncut square traces to $t_i^{(2)}(\sigma)$, and a bounded first
moment $\lim_\omega\tau_n(h_n(s_i))<\infty$.
There is no loss of square mass on regularization. For $x\geq0$
and $t>0$, scalar functional calculus gives
\begin{equation}\label{tc:square-mass}
0\leq\tau(x^2)-\tau((x^2-t)_+)
\leq\sqrt t\,\tau(x)
\end{equation}
whenever the right side is finite. Apply this with $x=h_n(s_i)$,
take the ultralimit, and then let $t\downarrow0$. The definition of
$\rho$ and \eqref{tc:square-mass} yield exactly
\eqref{tc:square}.
\smallskip\noindent\emph{Retain the lower ideal support.}
Fix one lower datum $(y,\eps,\eta)$ for the test $s=f\otimes a$.
For all sufficiently large $n$, \Cref{tc:rank-average} gives
\[
d_\tau((y-\eps)_+)
\leq C\,d_\tau((h_n(s)-\eta)_+)
\quad\text{for every }\tau\in T(Q),
\qquad C=2\|y\|^2/\eps^2.
\]
The order determination in \Cref{st:realification} implies
\[
[(y-\eps)_+]\leq
\lceil C\rceil[(h_n(s)-\eta)_+].
\]
If $(y-\eps)_+=0$, this comparison has no content and needs no
witness. Otherwise $C>0$; set $N=\lceil C\rceil\geq1$.
The uniform comparison implementation in
\Cref{seq:uniform-witness} gives rectangular columns $w_n$ such that
\[
\left\|w_n^*\operatorname{diag}_N(h_n(s))w_n-(y-\eps)_+\right\|
<1/n,
\qquad
\|w_n\|^2\leq\frac{\|y\|+1/n}{\eta}.
\]
This bound is independent even of $N$. The columns pass to the
sequence algebra and put $(y-\eps)_+$ in $\Ideal_D(h(s))$.
Letting the source cut decrease to zero puts $y$ itself in this ideal.
The previously chosen $y$'s and
their cuts generate $Q(U_a)$, proving \eqref{tc:lower-ideal}.
\end{proof}
==== END The-trace-cone-classifies-Razak-Jacelon-stabilizations//sections/06-tracial-cone.tex ====
==== BEGIN The-trace-cone-classifies-Razak-Jacelon-stabilizations//sections/07-cone-correction.tex ====
\section{Correction of the cone models}
\label{cc:section}
Throughout this section,
\[
S=C_0((0,1])\otimes P,\qquad D=Q_\infty.
\]
We first correct the multiplicative defects of the map in
\Cref{tc:rough-model}. The correction takes place on a hereditary
support containing the defects and a small cone representation.
Its ideal estimates are stronger than equivariance for the projected
labels. This strength is what permits the cone homotopy to be used
with the finer labels of $\Prim(S)$.
\subsection{The error support and its ideal estimates}
\begin{lemma}[A hereditary support for the defects]\label{cc:defect-support}
Let $h:S\to D$ be \cpc and suppose that, for every open $U\subset X$,
\begin{equation}\label{cc:mixed-defects}
h(sz)-h(s)h(z),\quad h(zs)-h(z)h(s)\in J(U)
\qquad(s\in S(U),\ z\in S).
\end{equation}
Let $E=C^*(h(S))$ and let $I\lhd E$ be the ideal generated by
$h(ab)-h(a)h(b)$, for $a,b\in S$. Then
\begin{equation}\label{cc:annihilation}
h(S(U))I+Ih(S(U))\subset J(U).
\end{equation}
There is a locally small cone homomorphism $\theta:S\to D$ such that,
after placing $h$ and $\theta$ in separate matrix corners, a
$\sigma$-unital hereditary algebra $H\subset M_2(D)$ has the following
properties. The first-corner map $h$ idealizes $H$, its multiplicative
defects belong to $H$, and $\theta(S)\subset H$. Moreover, for every
$x\in S_+$,
\begin{equation}\label{cc:strong-support}
h(S(U_x))H+Hh(S(U_x))\subset H_x,
\qquad H_x:=\Ideal_H(\theta(x)).
\end{equation}
Here $\theta(S(U))\subset J(U)$, with the matrix conventions of
\Cref{st:folding}.
\end{lemma}
\begin{proof}
Fix $U$ and pass to $D/J(U)$, writing bars for the quotient maps.
For $s\in S(U)$ and $a,b\in S$, the mixed multiplication identities give
\[
\bar h(s)\bigl(\bar h(ab)-\bar h(a)\bar h(b)\bigr)
=\bar h(sab)-\bar h(sa)\bar h(b)=0.
\]
The analogous right product is zero. More generally a word in
$\bar h(S)$ between $\bar h(s)$ and a defect can be absorbed into
the localized input one factor at a time. The resulting input is
still in $S(U)$. Polynomial approximation in $E$ therefore proves
\eqref{cc:annihilation} for the whole generated ideal $I$.
The algebra $E$, and hence $I$, is separable. Apply
\Cref{sm:coefficient-domination} to the bounded bilinear map
$(s,b)\mapsto h(s)b$, with $b\in I$. Include its adjoints in the
same countable family. This gives a locally small cone homomorphism
$\theta$ satisfying, simultaneously for every $x\in S_+$,
\begin{equation}\label{cc:ambient-domination}
h(S(U_x))I+Ih(S(U_x))
\subset\Ideal_{M_2(D)}(\theta(x)).
\end{equation}
The first-corner and second-corner placements are understood in this
formula; the ideal on the right is the ambient matrix ideal.
Choose strictly positive elements $e\in I_+$ and
$k\in\theta(S)_+$, omitting a zero summand when necessary, and set
\begin{equation}\label{cc:H-definition}
H=\Her_{M_2(D)}(e\oplus k).
\end{equation}
It is $\sigma$-unital, since $e\oplus k$ is strictly positive for $H$.
No separability assertion about $H$ is needed. Since $I\lhd E$,
$h(a)I+Ih(a)\subset I$ for $a\in S$. Functional-calculus
approximate units of $I$ show that the first-corner map idealizes
the first support. They also show that it preserves all the
off-diagonal corners in \eqref{cc:H-definition}: on the dense
linear span of products
\[
f(e\oplus k)\,z\,g(e\oplus k),\qquad z\in M_2(D),
\]
left multiplication by $h(a)$ replaces the first support factor by
an element of $I$, and right multiplication has the corresponding
property. Approximation by the support approximate unit proves
idealization of $H$. The defects themselves lie in $I\subset H$.
For $a\in S(U_x)$, the same calculation, now using
\eqref{cc:ambient-domination}, puts $h(a)H$ and $Hh(a)$ in
$H\cap\Ideal_{M_2(D)}(\theta(x))$. The hereditary ideal
correspondence gives
\[
H\cap\Ideal_{M_2(D)}(\theta(x))
=\Ideal_H(\theta(x)).
\]
This proves \eqref{cc:strong-support}; equivalently, one may apply
\Cref{un:hereditary-support}. All matrix placements preserve the
local smallness of $\theta$.
\end{proof}
We use the right Hilbert $H$-module convention. Write
$\mathcal L_H(\mathcal E)$ and $\mathcal K_H(\mathcal E)$ for
adjointable and compact operators on a Hilbert $H$-module $\mathcal E$.
For an ideal $L\lhd H$, let $\mathcal E L$ denote the closed span
of $\{\xi b:\xi\in\mathcal E, b\in L\}$.
\begin{lemma}[A dilation compatible with fine ideal labels]\label{cc:scaled-dilation}
Let $d:S\to\mathcal L_H(H)=M(H)$ be the multiplier map induced by
$h$ in \Cref{cc:defect-support}. There is a countably generated
Hilbert $H$-module $\mathcal E'$ and a representation
\begin{equation}\label{cc:dilation}
\Pi:S\longrightarrow\mathcal L_H(H\oplus\mathcal E'),
\qquad
\Pi(a)=
\begin{pmatrix}
d(a)&C(a)\\ B(a)&d'(a)
\end{pmatrix},
\end{equation}
whose off-diagonal entries are compact. The maps $d,d'$ are c.p.c.
The module may be stabilized, retaining the first copy of $H$, so
that $\mathcal E'$ is the standard countable module.
Give $S$ its tight $\Prim(S)$-labels and give $H$ the order-preserving
assignment
\begin{equation}\label{cc:fine-labels}
H(O)=\Ideal_H(\theta(S(O)))
\qquad(O\subset\Prim(S)\text{ open}).
\end{equation}
Let $\alpha_t$ be cone scaling,
\[
(\alpha_t a)(r)=a(tr),\qquad 0\le t\le1,
\]
where $a(0)=0$. Every $\Pi_t:=\Pi\alpha_t$ is weakly
$\Prim(S)$-nuclear for \eqref{cc:fine-labels}. The same is true
of its diagonal compressions $d_t,d'_t$.
\end{lemma}
\begin{proof}
Use the Hilbert-module Stinespring construction in
\cite[proof of Theorem~3, pp.~140--141]{Kasparov}, applied to the
unital extension $d^+$ of $d$ on the forced unitization of $S$.
The source is separable and $H$ has a countable approximate unit,
as required there.
The resulting representation acts on the completion of the
algebraic Stinespring module. Its adjointable isometry
$V:H\to\mathcal E$ is given by
$V\xi=1\otimes\xi$ and
$V^*(b\otimes\eta)=d^+(b)\eta$, so
$V^*\Pi(a)V=d(a)$. The module is countably generated:
use a countable dense subset of the unitized source and a countable
approximate unit of $H$. Decompose
$\mathcal E=VH\oplus\mathcal E'$ and identify $VH$ with $H$.
For $B(a)=(1-VV^*)\Pi(a)V$,
\begin{equation}\label{cc:compact-offdiagonal}
B(a)^*B(a)=d(a^*a)-d(a^*)d(a)\in H
=\mathcal K_H(H).
\end{equation}
An adjointable operator whose square modulus is compact is compact,
by functional calculus in the compact-operator ideal. Thus $B(a)$
is compact, and so is $C(a)=B(a^*)^*$. Kasparov stabilization
\cite[Theorem~2]{Kasparov} of $\mathcal E'$ after adding a zero
representation identifies its
enlargement with the standard module. This leaves $VH$ fixed and
preserves compactness and the indicated diagonal compressions.
We check the ideal assertion before using the homotopy. For
$a\in S(U_x)$, coefficients on the dense Stinespring generators
have the form
\begin{equation}\label{cc:Stinespring-coefficients}
\langle b\otimes\xi,\Pi(a)(c\otimes\eta)\rangle
=\xi^*d(b^*ac)\eta\in H_x,
\end{equation}
where $b,c$ belong to the forced unitization. The containment uses
$b^*ac\in S(U_x)$ and \eqref{cc:strong-support}. It extends to
all vectors. Passing to the quotient module by $\mathcal E H_x$
shows that $\Pi(a)\mathcal E\subset\mathcal E H_x$; the same
holds for the adjoint. Stabilization and module unitaries preserve
these ideal submodules.
Now take $a\in S(O)$ and put $x=|a|$. Scaling does not enlarge its
\emph{projected} support:
\begin{equation}\label{cc:projected-scaling}
\alpha_t(a)\in S(U_x)\quad(0\le t\le1).
\end{equation}
Indeed a point quotient in the $P$ coordinate which kills $a$
also kills every rescaling of $a$. Since $x\in S(O)$,
$H_x\subset H(O)$, and \eqref{cc:Stinespring-coefficients}
applied to $\alpha_t(a)$ proves weak equivariance of $\Pi_t$.
It proves the same assertion for the compressions. Source ideals
and quotients are nuclear, so these coefficient conditions also
give weak nuclearity, as in \Cref{un:absorption}.
This argument does not assert that $\alpha_t$ preserves $S(O)$.
The stronger estimate for the entire projected ideal
$S(U_x)$ is the reason that the scaled representation has the
required fine ideal bounds.
\end{proof}
For later use, if $U\subset X$ and $O_U=(0,1]\times U$, then
\begin{equation}\label{cc:local-compact-ideal}
H(O_U)\subset H\cap M_2(J(U)).
\end{equation}
This follows from the ideal property of $J(U)$ and the local
smallness of $\theta$. If a compact-valued map is the difference
of two weakly equivariant maps, its value on $S(O_U)$ belongs to
the compact operators over $\mathcal E H(O_U)$. To see this,
pass to the quotient module by $\mathcal E H(O_U)$: both maps
vanish there on this input, and the kernel of the quotient map on
compact operators is precisely
$\mathcal K_H(\mathcal E H(O_U))$. Thus every such compact
correction is locally in $J(U)$.
\subsection{A finite correction on the original module}
The dilation now respects every fine ideal label, including along
cone scaling. We will compare nearby times in a finite subdivision of
that homotopy. Absorption makes a finite chain of dilations available as
a compact perturbation of a repeat of $\theta$. Shifting the first
summands along this chain pairs each complementary summand with a nearby
compression; the unpaired summand lies near the zero endpoint. The
resulting map is close to a representation.
We then truncate only the compact perturbation. This produces a finite
matrix correction with the original $h$ still in its first corner, so
its square moments and lower ideal support remain available.
\begin{lemma}[A finite correction retaining the original corner]\label{cc:finite-correction}
Under the hypotheses of \Cref{cc:defect-support}, let
$\mathcal F$ be a finite subset of the unit ball of $S$ and let
$\eta>0$. There are an integer $N$, a bounded linear
$*$-preserving map $L:S\to M_{N+1}(M_2(D))$, and a bounded linear
map $K_N:S\to M_{N+1}(H)$ such that
\begin{equation}\label{cc:finite-form}
L(a)=h(a)\oplus\theta(a)^{\oplus N}+K_N(a),
\end{equation}
and, for $a,b\in\mathcal F$,
\begin{equation}\label{cc:finite-estimates}
\|L(a)\|\le\|a\|+\eta,
\qquad
\|L(ab)-L(a)L(b)\|\le3\eta+2\eta^2.
\end{equation}
For every open $U\subset X$,
\begin{equation}\label{cc:finite-locality}
K_N(S(U))\subset M_{N+1}(H\cap M_2(J(U))).
\end{equation}
The first summand in \eqref{cc:finite-form} is the original $h$,
with only zero corner padding.
\end{lemma}
\begin{proof}
\smallskip\noindent\emph{Absorb a finite chain of dilations.}
Include $\mathcal F^*$ and all products from $\mathcal F$ in the
finite list on which estimates will be imposed. By norm continuity
of $t\mapsto\alpha_t(a)$, choose
\[
0=t_0<t_1<\cdots<t_k=1
\]
so that adjacent values differ in norm by less than $\eta$ on
this enlarged list. Write $d_i=d_{t_i}$, $d'_i=d'_{t_i}$ and
$\Pi_i=\Pi_{t_i}$, using the same first module $H$ and the same
complementary module in all these dilations.
For the absorption application put
$\widehat\theta(a)=\theta(a)\otimes e_{11}\in H\otimes\K$,
with target labels $H(O)\otimes\K$.
If $O_a$ is the fine support of $a\in S_+$, then
$S(O_a)=\Ideal_S(a)$ and
\[
\Ideal_H(\theta(S(O_a)))=\Ideal_H(\theta(a)).
\]
Thus $\widehat\theta$ is full for these fine labels, independently
of any coarse fullness assertion. Apply \Cref{un:absorption}
to a literal stable infinite repeat $\Theta$ of
$\widehat\theta$ and the finite sum of the $\Pi_i$.
On the standard Hilbert $H$-module, this repeat can be reindexed as
\begin{equation}\label{cc:stable-repeat}
\Theta(a)=\operatorname{diag}
(\theta(a),0,\theta(a),0,\ldots).
\end{equation}
Indeed each repeated copy of $\widehat\theta$ acts by $\theta$
on one $H$ slot and by zero on the other slots of its standard
module. Both the active slots and the remaining zero slots are
countably infinite, giving \eqref{cc:stable-repeat} by a fixed
permutation. We retain every zero slot throughout absorption and
finite compression. The bare diagonal repeat with all these zero
slots removed need not absorb a zero representation.
The hypotheses hold: the tightly labeled source is separable,
exact and lower semicontinuous; $H\otimes\K$ is $\sigma$-unital
and stable; and \Cref{cc:scaled-dilation} supplies weak nuclearity
of the representations being absorbed. Absorption identifies
the repeat module with a module carrying
\[
\Theta\oplus\Pi_1\oplus\cdots\oplus\Pi_k
\]
so that its difference from the original repeat is compact on
every input. More precisely, absorption in
\cite[Definition~5.4]{Gabe} gives a repeat-side module unitary $U$
with
\[
U^*(\Theta\oplus\textstyle\bigoplus_i\Pi_i)(a)U-\Theta(a)
\in\mathcal K_H(H^{\oplus\infty})\qquad(a\in S);
\]
the compact membership holds for every input for each chosen
unitary, in addition to the finite-test norm approximation.
Only this compact-membership conclusion from absorption is needed
below. Its finite-test approximation error does not enter our bound:
after the unitary identification we keep the absorbed representation
itself. All these identifications act on the repeat side; the initial
module carrying $d=d_k$ is fixed.
\smallskip\noindent\emph{Shift the first summands along the chain.}
Denote the initial copy of
$H$ by $H_0$, the first copy in $\Pi_i$ by $H_i$, and its
complement by $\mathcal E'_i$. Delete the compact off-diagonals
of the absorbed $\Pi_i$. Apart from the unused repeat, the
diagonal summands are
\begin{equation}\label{cc:before-regrouping}
d_k\big|_{H_0}\ \oplus
\bigoplus_{i=1}^k
\left(d_i\big|_{H_i}\oplus d'_i\big|_{\mathcal E'_i}\right).
\end{equation}
Pair $H_0$ with $\mathcal E'_k$, and, for $i<k$, pair
$H_{i+1}$ with $\mathcal E'_i$. The unused first module is
$H_1$. Restore the compact off-diagonals of $\Pi_i$ in the pair
whose second component is $\mathcal E'_i$. In this ordered
decomposition the resulting map is
\begin{equation}\label{cc:regrouped-map}
T'=\Theta\oplus d_1\oplus
\bigoplus_{i=1}^{k-1}
\left[\Pi_i+
\begin{pmatrix}d_{i+1}-d_i&0\\0&0\end{pmatrix}\right]
\oplus\Pi_k.
\end{equation}
It is a compact perturbation of the map in
\eqref{cc:before-regrouping}, including its repeat: the diagonal
entries were only permuted, and the restored entries are compact.
The representation
\begin{equation}\label{cc:comparison-representation}
R'=\Theta\oplus0\oplus
\bigoplus_{i=1}^{k-1}\Pi_i\oplus\Pi_k
\end{equation}
is within $\eta$ of $T'$ on the enlarged finite list. Indeed
$\|d_1(a)\|\le\|\alpha_{t_1}(a)\|<\eta$ there, since
$\alpha_0=0$, and every other difference in
\eqref{cc:regrouped-map} has norm less than $\eta$.
No compactness of $d_{i+1}(a)-d_i(a)$ has been used.
Pull both maps back through the block permutation and the
absorption identification to the original module
\[
\mathcal L=H\oplus H^{\oplus\infty}.
\]
Writing $T_0=d\oplus\Theta$, we obtain
\begin{equation}\label{cc:T-plus-K}
T=T_0+K,\qquad
K(a)\in\mathcal K_H(\mathcal L),\qquad
\|T(a)-R(a)\|<\eta
\end{equation}
on the enlarged list, where $R$ is a representation.
The maps in this construction are bounded linear and
$*$-preserving. They are weakly equivariant: the block
compressions and off-diagonal entries have the ideal bounds of
\Cref{cc:scaled-dilation}, and any Hilbert-module unitary preserves
the submodule associated to an ideal. Thus
\eqref{cc:local-compact-ideal} applies to $K$.
For $a,b\in\mathcal F$, comparison with the contractive
representation $R$ gives
\begin{equation}\label{cc:T-estimates}
\|T(a)\|\le\|a\|+\eta,
\qquad
\|T(ab)-T(a)T(b)\|\le3\eta+\eta^2.
\end{equation}
\smallskip\noindent\emph{Compress the compact correction.}
Let $E_N$ be the projection onto the initial copy of $H$ and the
first $N$ slots in the \emph{original} diagonal repeat
\eqref{cc:stable-repeat}, including both active and zero slots. It
commutes with $T_0(a)$ for all $a$. The projections $E_N$ tend
strictly to the identity on the compact operators. Choose $N$
so that both tails of every $K(a)$ on the enlarged list have
norm less than $\eta$. Then
\begin{align}
&E_NT(ab)E_N-E_NT(a)E_NT(b)E_N\notag\\
&\quad=E_N\bigl(T(ab)-T(a)T(b)\bigr)E_N
+E_NK(a)(1-E_N)K(b)E_N.
\label{cc:compression-defect}
\end{align}
The second term has norm at most $\eta^2$. Compression preserves
the norm estimate in \eqref{cc:T-estimates}.
The entries of $E_NK(a)E_N$ lie in $H$, so the compressed map
has an actual ambient realization with diagonal baseline $h$
followed by the retained $\theta$ and zero slots. To obtain
exactly \eqref{cc:finite-form}, let $Z_N(a)$ be the finite diagonal
matrix which places $\theta(a)$ in each retained zero repeat slot
and zero in every other slot, including the initial one. Define
\begin{equation}\label{cc:zero-slot-rebasing}
K_N(a)=E_NK(a)E_N-Z_N(a).
\end{equation}
Replacing each zero baseline slot by $\theta(a)$ and subtracting
it in \eqref{cc:zero-slot-rebasing} leaves the compressed map $L$
unchanged. The map $Z_N$ is bounded linear and $*$-preserving,
has entries in $H$, and is locally in $J(U)$ because $\theta$ is
locally small. Thus $K_N$ has the required compact and local
properties. This rebasing is performed only after finite
compression; the tail estimate used the original $T_0$ with
all its zero slots.
\smallskip\noindent\emph{Return to the ambient sequence algebra.}
To transfer the multiplier estimates, let $p_H$ be the support
projection of $H$ in $M_2(D)^{**}$. Idealization implies that
$h(a)$ commutes with $p_H$. The correction entries, and the
values of $\theta$, are supported on $p_H$. On its matrix
amplification the ambient map is exactly the compressed
multiplier map just estimated. On the orthogonal complement
all corrections vanish; the remaining first-corner map $h$
is contractive and multiplicative there, since its defects lie
in $I\subset H$. Taking the maximum of these two reducing
parts proves \eqref{cc:finite-estimates} in the ambient norm.
Finally \eqref{cc:finite-locality} follows from the local
compact ideal statement for $K$.
\end{proof}
\subsection{Passage to a corrected sequence homomorphism}
\begin{proposition}[Cone correction]\label{cc:correction}
Suppose $h:S\to D$ is \cpc and satisfies
\eqref{cc:mixed-defects}. After trace-preserving coordinate
corner padding, there is a homomorphism $\lambda:S\to D$ such
that
\begin{equation}\label{cc:corrected-difference}
(\lambda-h^{\mathrm{pad}})(S(U))\subset J(U)
\qquad(U\subset X\text{ open}).
\end{equation}
The padding can be chosen to fix the constant copy of $Q$ in
the sequence quotient. Consequently it preserves the constant
ideal comparisons and the regularized trace formulas prescribed
for $h$.
\end{proposition}
\begin{proof}
If the defect ideal is zero, $h$ is already a homomorphism.
Otherwise use the preceding construction. Nuclearity and separability
of $S$ provide a \cpc coordinate lift of $h$ by
\cite[Theorem~3.10]{ChoiEffrosLifting}; fix one such lift.
Fix a countable
rational $*$-subalgebra dense in $S$, together with countable
dense local tests in $S(V)$ for a compact-containment basis and
all finite unions of its members. Include rational linear
combinations, products, adjoints, and positive cuts needed for
the algebraic and ideal tests. Fix $\eta_j\downarrow0$. At stage $j$
rescale the first $j$ tests and their relevant products into the unit
ball. Apply \Cref{cc:finite-correction} with a tolerance small enough
that, after rescaling back, its norm and multiplicativity errors on
these tests are at most $\eta_j$. Denote the resulting map by $L_j$
and its finite repeat size by $N_j$. In particular,
$\|L_j(a)\|\leq\|a\|+\eta_j$ for every current input $a$.
Write $h^{\mathrm{pad},j}=h\oplus0^{\oplus N_j}$ in the current
row, and subtract it from $L_j$. The remaining terms
are its compact correction and the $N_j$ small copies of
$\theta$. They belong to the appropriate matrix ideal $J(V)$
on every tested input in $S(V)$. On each fixed input $a$, these row
differences also satisfy
\[
\|L_j(a)-h^{\mathrm{pad},j}(a)\|
\leq2\|a\|+\eta_j
\]
for all sufficiently large $j$. Thus they meet the boundedness
hypothesis for prescribed errors in \Cref{seq:diagonal}; no uniform
bound on the global norms of the maps $L_j$ is needed.
For each fixed positive cut
tolerance their row estimates therefore have the form
\begin{equation}\label{cc:row-smallness}
d_\tau\bigl((|z_{j,n}|-\eps)_+\bigr)
\le r_{j,n}\,d_\tau(c_j),\qquad
r_{j,n}\longrightarrow0\quad(n\to\infty),
\end{equation}
where $c_j$ is compactly Cuntz supported in $Q(V)$ and the
bound is uniform in $\tau$.
For each scheduled pair $V\Subset U$, first fix a cut-positive
$b_{V,U}\in Q(U)$ whose ideal contains $Q(V)$. Compact support
gives $[c_j]\le M_j[b_{V,U}]$ for a finite integer $M_j$.
At stage $j$, wait in its coordinate row until
$M_jr_{j,n}\le 1/j$ for all the finitely many current tests and
cut tolerances. Wait also until their coordinate norm,
involution and product estimates agree with the rescaled finite-test
bounds, with an additional error tending to zero. The integers $M_j$
and $N_j$ are chosen before this waiting time; neither needs a bound independent of $j$.
This is the diagonal procedure of \Cref{seq:diagonal}.
Choose coordinate representatives linearly on each finite
rational span, preserving the representatives already needed
for its displayed algebraic tests. The preceding eventual
estimates allow successive rows to be joined. Matrix folding
in \Cref{st:folding} is applied only after these choices. Its
corner identification is adjusted to be within $1/j$ of the
identity on the first $j$ constants of a fixed dense subset of
$Q$. The resulting corner map on the sequence algebra fixes
$Q$ pointwise. Apply the corner embeddings to the fixed coordinate
lift of $h$ to define
$h^{\mathrm{pad}}$.
The joined corrected representatives define a contractive,
linear, $*$-preserving, multiplicative map on the dense
rational $*$-subalgebra: each asserted identity and norm bound
is one of the eventually imposed tests. It extends uniquely
to a homomorphism $\lambda:S\to D$. The joined estimates
\eqref{cc:row-smallness}, now against the fixed
$b_{V,U}$, put its difference from $h^{\mathrm{pad}}$ in
$J(U)$ on the tests from $S(V)$. Both resulting maps are
continuous. Since the ideals $S(V)$ with $V\Subset U$ span
densely in $S(U)$ and $J(U)$ is closed, this proves
\eqref{cc:corrected-difference} for every open $U$.
For completeness, if $\iota_n:Q\to Q$ are the coordinate
corner embeddings, they induce a homomorphism $\iota:D\to D$
with $\iota|_Q=\id_Q$ and $h^{\mathrm{pad}}=\iota h$.
The pullback of a regularized trace along $\iota$ is the
regularized trace of the coordinate weights
$\tau_n\iota_n$, with the same restriction $\sigma$ to $Q$.
Thus every prescribed formula for $h$ of this kind is
preserved. Applying $\iota$ to an ideal comparison involving
constants also preserves that comparison and those constants.
\end{proof}
\subsection{Recovery of the full trace formula}
We record the trace determination used at the last step. A
positive element compactly Cuntz supported in the densely
finite ideal of a lower-semicontinuous trace has finite rank.
Indeed that ideal is generated by finite-trace positives;
their positive cuts have finite rank. Compact containment
dominates the given element by finitely many such cuts.
Conversely, if an ideal is generated by finite-rank positives,
the trace is densely finite on it, since their hereditary
algebras consist of finite-trace elements.
\begin{lemma}[Determination on a finite ideal]\label{cc:finite-trace-determination}
Let $V\subset X$ be open. Suppose two lower-semicontinuous
traces on $S(V)$ agree on all positive even integer powers of the rich
elementary positive test supply of \Cref{tc:rough-model}, and
these values are finite. If the cuts of that supply generate
$S(V)$, the two traces are densely finite and equal on $S(V)$.
\end{lemma}
\begin{proof}
Write $\omega_1,\omega_2$ for the traces, and $\omega$ for
either one when making an estimate.
Finite second moments give finite ranks on all positive cuts
of a test $s$: for $\eps>0$,
\[
d_\omega((s-\eps)_+)\le\eps^{-2}\omega(s^2).
\]
The preceding observation proves dense finiteness for both
traces. Fix $s$ and a continuous function $g$ on its squared
spectrum which vanishes near zero. Approximate $g(t)/t$
uniformly by polynomials. Equality of the moments
$\omega(s^{2k})$ therefore gives equality on $g(s^2)$:
the error is bounded by the uniform polynomial error times
$\omega(s^2)$. Increasing such functions to spectral support
indicators gives equality of the ranks of every positive cut
of $s^2$.
The test supply contains cuts from dense sets of the two
factors, so these squares approximate every positive
elementary tensor in $S(V)$. The comparison
\[
\|a-b\|<\delta
\quad\Longrightarrow\quad
(a-t-\delta)_+\precsim(b-t)_+
\qquad(t\ge0)
\]
transfers equality of all cut ranks to each such elementary
limit, by allowing the cut slack to decrease to zero. Spectral
integration now gives equality of trace values on all positive
elementary tensors.
To pass from elementary tensors to the whole algebra, let $L$ be a
relatively compact open subset of $(0,1]$ and let
$H_c=\Her(c)\subset P(V)$, where $c$ is compactly Cuntz supported
in $P(V)$. Choose $f_0\in C_c((0,1])_+$ equal to one on
$\overline L$, and write $c\precsim(b-\delta)_+$ with
$b\in P(V)_+$ and $\delta>0$. Then
\[
f_0\otimes c\precsim(f_1\otimes b-\delta/2)_+
\]
for a compactly supported $f_1$ equal to one on $\supp(f_0)$.
The larger tensor belongs to the common finite ideal $S(V)$ already
established above. Thus $f_0\otimes c$ has finite rank for both
traces. The support of $C_0(L)\otimes H_c$ is below its support,
so both traces are bounded on this localized algebra, with norms
bounded by that rank. A positive continuous $P$-valued
function there is uniformly approximated by finite positive
elementary sums, using a scalar partition of unity.
Additivity and boundedness yield equality on this localized
algebra. Finally take increasing scalar cutoffs and an
increasing cut-supported approximate unit of $P(V)$.
Traciality writes the corresponding compressions as
increasing positive approximations under the square root of
the element being tested. Lower semicontinuity extends the
equality to every positive element of $S(V)$.
\end{proof}
\begin{theorem}[Existence of cone models]\label{cc:cone-existence}
Let $m$ be a finite full-support Radon measure on $(0,1]$.
There is a homomorphism $p:S\to D$ such that, for every
regularized limit trace $\rho$ with $\sigma=\rho|_Q$,
\begin{equation}\label{cc:full-cone-formula}
\rho\circ p=m\otimes F(\sigma).
\end{equation}
This includes weights whose densely finite ideal is proper
or zero.
\end{theorem}
\begin{proof}
We first preserve the lower ideal signal through correction. For a fixed
weight we will then identify its trace on the prescribed finite ideal,
and use that signal to rule out any larger finite ideal.
\smallskip\noindent\emph{Preserve the lower signal.}
Take $h$ from \Cref{tc:rough-model} and correct it by
\Cref{cc:correction}. For each controlled positive test
$s=f\otimes a$, put
$\delta(s)=\lambda(s)-h^{\mathrm{pad}}(s)$.
It belongs to $J(U_s)$. Apply the cone version of
\Cref{sm:stabilizers} once more to obtain a locally small
homomorphism $\theta_1:S\to D$ with
\begin{equation}\label{cc:second-stabilizer}
\delta(s)\in\Ideal_D(\theta_1(s))
\end{equation}
for every such test. The countable coefficient form permits
all the required cuts and localized families simultaneously.
Set $p=\lambda\oplus\theta_1$ and fold the two corners,
again fixing the constants. In the following comparisons
we retain matrix notation for this folding.
The ideal of $p(s)$ contains both $\lambda(s)$ and
$\theta_1(s)$, hence contains $h^{\mathrm{pad}}(s)$ by
\eqref{cc:second-stabilizer}. The lower signal of
\Cref{tc:rough-model} and the preservation of constants give
\begin{equation}\label{cc:retained-signal}
Q(U_a)\subset\Ideal_D(p(f\otimes a))
\qquad(f\ne0)
\end{equation}
for each controlled test. Also
\begin{equation}\label{cc:total-smallness}
(p-h^{\mathrm{pad}})(S(U))\subset J(U),
\end{equation}
where zero padding of $h^{\mathrm{pad}}$ is understood.
\smallskip\noindent\emph{Identify the trace on its prescribed finite ideal.}
Fix a regularized trace $\rho$ and write
$Q(V)=\Fin(\sigma)$. By \Cref{st:ideals},
$\Fin(F\sigma)=P(V)$. Put
\[
\omega=\rho\circ p,\qquad \nu=m\otimes F\sigma.
\]
The trace $\rho$ vanishes on $J(V)$ by
\Cref{seq:smallideal-properties}. It therefore factors
through the quotient by this zero ideal. For each controlled
$s=f\otimes a$ localized with its controller in $V$,
\Cref{tc:rough-model} and \eqref{cc:total-smallness} give
\begin{equation}\label{cc:square-agreement}
\omega(s^2)
=\rho\bigl(h^{\mathrm{pad}}(s)^2\bigr)
=\int f^2\,dm\, F\sigma(a^2)
=\nu(s^2)<\infty.
\end{equation}
Apply the same formula to the positive powers in the supply.
The cuts of these localized elementary tests generate
$S(V)$, including when $V$ is obtained as an increasing union
of the basis opens. Thus \Cref{cc:finite-trace-determination}
proves
\begin{equation}\label{cc:finite-region}
S(V)\subset\Fin(\omega),\qquad
\omega|_{S(V)}=\nu|_{S(V)}.
\end{equation}
\smallskip\noindent\emph{Exclude an additional finite region.}
We show that the first inclusion is equality. Suppose the
open set corresponding to $\Fin(\omega)$ contains a
rectangle $L\times U$ not contained in $(0,1]\times V$.
Use a scalar test with compact support in $L$ and a cut
positive $a$ from the basis supply in $P(U)$. The tests may
be chosen with slightly enlarged supports still in this
rectangle. Their tensor products are then compactly Cuntz
supported in $\Fin(\omega)$, and consequently have finite
$\omega$-rank. For any such nonzero scalar test, write
$s=f\otimes a$. If $c\in Q(U_a)_+$ and $\eps>0$,
\eqref{cc:retained-signal} gives, for suitable finite $M$
and $\eta>0$,
\begin{equation}\label{cc:constant-finiteness}
[(c-\eps)_+]\le M[(p(s)-\eta)_+]
\quad\hbox{in }\Cu(D).
\end{equation}
It follows that
\[
d_\sigma((c-\eps)_+)
=d_\rho((c-\eps)_+)
\le M d_\omega(s)<\infty.
\]
Hence $\sigma$ is densely finite on $Q(U_a)$, so
$U_a\subset V$. The supports $U_a$ of these cut tests
exhaust $U$. This implies $U\subset V$, contradicting the
choice of the rectangle. Therefore
\begin{equation}\label{cc:exact-finite-ideal}
\Fin(\omega)=S(V)=\Fin(\nu).
\end{equation}
Both traces take value infinity on positive elements outside
this ideal. Together with \eqref{cc:finite-region}, this
proves \eqref{cc:full-cone-formula}. The argument also
applies when $V$ is empty: there is no finite region to
identify, and the retained signal excludes every nonzero
additional finite ideal.
\end{proof}
==== END The-trace-cone-classifies-Razak-Jacelon-stabilizations//sections/07-cone-correction.tex ====
==== BEGIN The-trace-cone-classifies-Razak-Jacelon-stabilizations//sections/08-point-intertwining.tex ====
\section{Point models and approximate intertwining}\label{pt:section}
We now construct homomorphisms \(P\to Q\) and \(Q\to P\),
and then an isomorphism realizing the prescribed invariant.
Cone existence first gives a model on the line. Norm uniqueness
makes translation covariant, so the model extends to a crossed
product. A trace-normalized projection then gives a point model.
A second application of uniqueness makes its late coordinate
lifts close modulo unitaries, yielding an actual homomorphism.
The final application makes the two compositions approximate
identities and completes the intertwining.
The suspension, crossed-product projection, and subsequence
extraction also occur in \cite[Remark~6.4]{GabeO2}. The suspension,
crossed-product, and full-corner strategy adapts the proof of
\cite[Theorem~14.1]{Gabe}. The latter theorem has an
\(\mathcal O_\infty\)-stable target and is not applied directly here.
Here each passage must preserve the
full extended trace data. We compute the projection's trace by
an identity of finite positive sums, and recover the invariant
of the limiting maps through positive cuts.
We use \(\Ad(w)(a)=waw^*\).
\subsection{The line and its translation}
\begin{proposition}[A covariant line model]\label{pt:suspension}
There is a model
\[
\lambda:C_0(\R)\otimes P\longrightarrow D
\]
for Lebesgue measure and \(F\).
Let \(\alpha(f)(t)=f(t-1)\), acting trivially on \(P\).
For c.p.c. coordinate lifts \(\lambda_n\), there are
unitaries \(u_n\in M(Q)\) such that
\begin{equation}\label{pt:coordinate-covariance}
\|\Ad(u_n)(\lambda_n(s))-\lambda_n(\alpha(s))\|
\longrightarrow0\qquad(s\in C_0(\R)\otimes P).
\end{equation}
\end{proposition}
\begin{proof}
For \(j\geq2\), put \(I_j=(-j,j]\) and give it Lebesgue
measure. This is a finite full-support cone measure, so
\Cref{cc:cone-existence}, transported along a homeomorphism
\((0,1]\cong I_j\), gives a model
\(\lambda^{(j)}:C_0(I_j)\otimes P\to D\).
Choose \(0\leq\eta_j\leq1\) in \(C_c((-j,j))\), equal
to one on \([-j+1,j-1]\). The maps
\[
R_j:C_0(\R)\otimes P\longrightarrow C_0(I_j)\otimes P,
\qquad R_j(s)=\eta_j\,s|_{I_j},
\]
are c.p.c.: restriction followed by multiplication by the scalar
positive contraction $\eta_j$ is completely positive and contractive,
and its values vanish at the omitted left endpoint. The domain
$C_0(I_j)\otimes P$ of each $\lambda^{(j)}$ is separable and nuclear.
The maps $R_j$ are exactly multiplicative on every
fixed finite set of compactly supported inputs once \(j\)
is large enough to contain their supports in the region
where \(\eta_j=1\).
Take c.p.c. coordinate lifts of each \(\lambda^{(j)}\),
by \cite[Theorem~3.10]{ChoiEffrosLifting}, and compose with \(R_j\).
Apply \Cref{seq:diagonal} to an increasing dense list of
compactly supported algebra tests and the elementary rank
tests of \Cref{seq:elementary-tests}.
Here the tests are fixed on the line before increasing
\(j\): each function, its positive trims, and its enlarged
compact support all lie inside \((-j+1,j-1)\) in every
sufficiently late row. Their Lebesgue masses are therefore
unchanged by \(R_j\).
For every sandwich test choose the strict scalar slack,
the enlarged source, and the trimmed target before selecting
coordinates in that row. The constant Cuntz representatives
in \(Q\) are the same for all these rows.
The fixed target tolerances give the bounded comparison
witnesses of \Cref{seq:uniform-witness}; thus the row
diagonal retains the comparisons in \(D\), even when the
row thresholds vary.
Choose the coordinates for the first \(j\) tests only
after these choices, with algebra errors at most \(j^{-1}\).
The resulting c.p.c. sequence \(\lambda_n\) is
asymptotically multiplicative. Its quotient
\(\lambda\) satisfies the elementary rank identities for
every regularized limit, and
\Cref{seq:elementary-tests} proves the asserted model
identity.
Lebesgue measure is translation invariant, including the
extended product convention. Hence \(\lambda\) and
\(\lambda\circ\alpha\) are models for the same data.
Apply \Cref{un:uniqueness} to their specified coordinate
lifts to obtain \eqref{pt:coordinate-covariance}.
\end{proof}
The covariance unitaries belong to \(M(Q)\), whereas the coordinate model
values belong to \(Q\). To place them in one unital algebra, let
\[
\mathcal C=\prod_{n=1}^{\infty}M(Q)\Big/
\bigoplus_{n=1}^{\infty}M(Q).
\]
The product here consists of bounded sequences and the direct sum
of norm-null sequences. There is a natural faithful inclusion
\(D\subseteq\mathcal C\), and \(D\) is an ideal in
\(\mathcal C\).
The sequence from \Cref{pt:suspension} defines a unitary
\(u\in\mathcal C\), and the covariance relation is exact:
\(\Ad(u)\lambda(s)=\lambda(\alpha(s))\).
Integration of this covariant pair gives a homomorphism
\begin{equation}\label{pt:crossed-map}
\Psi:(C_0(\R)\otimes P)\rtimes_\alpha\Z
\longrightarrow D.
\end{equation}
This also applies when \(\lambda\) is degenerate: in a faithful
representation of \(\mathcal C\) on \(\mathcal H\), restrict to the reducing
subspace \(\overline{\lambda(C_0(\R)\otimes P)\mathcal H}\),
integrate there, and extend by zero. The finite Fourier sums
have the form \(\sum_n\lambda(s_n)u^n\), which lie in the
ideal \(D\); norm closure gives the stated target.
Because the action on \(P\) is trivial and \(\Z\) is
amenable, the domain is canonically
\[
E\otimes P,\qquad E=C_0(\R)\rtimes_\alpha\Z.
\]
The identification follows also by matching the commuting
representations of \(P\) and the covariant representation
of \(C_0(\R)\). Nuclearity makes the tensor-product norm
unambiguous. We denote the canonical implementing
multiplier of \(E\) by \(v\), so \(vf v^*=\alpha(f)\).
\subsection{An explicit trace-normalized corner}
For any projection \(p\in E\), the map \(a\mapsto p\otimes a\)
is a homomorphism from \(P\) to \(E\otimes P\). We seek a
projection for which composition with \(\Psi\) has exactly the
point trace data. The next lemma supplies the needed normalization
by a finite positive partition identity. It also identifies the
corresponding full corner of \(E\).
\begin{lemma}[The corner projection]\label{pt:corner}
There are \(\zeta\in C_c(\R)_+\) and a projection \(p\in E\)
such that
\[
\sum_{n\in\Z}\zeta(t-n)^2=1,
\qquad \int_\R\zeta(t)^2\,dt=1,
\]
and, for any \(C^*\)-algebra \(R_0\), any extended trace
\(\omega\) on \(E\otimes R_0\), and every \(a\in(R_0)_+\),
\begin{equation}\label{pt:corner-trace}
\omega(p\otimes a)=\omega(\zeta^2\otimes a).
\end{equation}
In particular, if the restriction of \(\omega\) to
\(C_0(\R)\otimes R_0\) is Lebesgue measure tensored with
\(\nu\), then \(\omega(p\otimes a)=\nu(a)\), including
the value infinity. Moreover, \(p\) is full and
\(pEp\cong C(\R/\Z)\).
\end{lemma}
\begin{proof}
Choose \(\eta\in C_c(\R)_+\) strictly positive on
\([0,1]\), and set
\[
H(t)=\sum_{n\in\Z}\eta(t-n)^2,\qquad
\zeta(t)=H(t)^{-1/2}\eta(t).
\]
The sum defining \(H\) is locally finite, positive,
continuous, and one-periodic. The partition identity
for \(\zeta\) follows immediately. Integrating that
identity over \([0,1]\) gives the asserted normalization.
Put
\begin{equation}\label{pt:projection-formula}
p=\sum_{n\in\Z}p_n v^n,
\qquad p_n(t)=\zeta(t)\zeta(t-n).
\end{equation}
Only finitely many coefficients are nonzero. The
involution formula gives
\(\alpha_n(p_{-n})=p_n\), so \(p^*=p\).
Convolution gives, for every \(k\in\Z\),
\[
(p^2)_k(t)
=\zeta(t)\zeta(t-k)\sum_{n\in\Z}\zeta(t-n)^2
=p_k(t).
\]
Thus \(p\) is a projection.
Choose finitely many real functions \(q_i\in C_c(\R)\),
each supported in an interval of diameter less than one,
such that
\[
q:=\sum_iq_i^2=1\quad\hbox{on }\supp(\zeta).
\]
For completeness, choose bump functions \(r_i\) in such
intervals whose squared sum is positive near
\(\supp(\zeta)\), and a cutoff \(\kappa\) equal to one
on that support and supported where this sum is positive.
Then \(q_i=\kappa r_i/(\sum_jr_j^2)^{1/2}\), extended
by zero, have the required properties.
The coefficient formula for \(p\) yields
\begin{equation}\label{pt:positive-partition}
qp=p=pq,\qquad \sum_iq_i p q_i=\zeta^2.
\end{equation}
To verify the second identity, its \(n\)-th coefficient
is
\[
\zeta(t)\zeta(t-n)\sum_iq_i(t)q_i(t-n).
\]
This vanishes when \(n\ne0\), because each support has
diameter less than one, and equals \(\zeta(t)^2\) when
\(n=0\).
For \(a\in(R_0)_+\), set \(y_i=q_i p\otimes a^{1/2}\).
These are elements of \(E\otimes R_0\), and
\eqref{pt:positive-partition} gives the exact positive
identities
\[
\sum_i y_i^*y_i=p\otimes a,\qquad
\sum_i y_i y_i^*=\zeta^2\otimes a.
\]
Additivity and \(\omega(y_i^*y_i)=\omega(y_i y_i^*)\)
prove \eqref{pt:corner-trace}. There are only finitely
many positive summands. In particular, this calculation
uses no subtraction of infinite values and requires no
finiteness assumption on \(\omega\).
For the Lebesgue trace assertion, the extended product formula
on the right has value
\((\int\zeta^2)\nu(a)=\nu(a)\).
We also verify fullness and the corner description.
Equation~\eqref{pt:positive-partition} puts
\(\zeta^2\) in \(\Ideal_E(p)\).
Conjugation by \(v^n\) preserves this ideal, so it contains
every \(\alpha_n(\zeta^2)\).
For \(f\in C_c(\R)\), the partition identity expresses
\[
f=\sum_{n\in\Z}f\,\alpha_n(\zeta^2)
\]
as a finite sum of elements of that ideal.
It therefore contains \(C_0(\R)\), and then the dense
finite Fourier sums in \(E\). Hence \(p\) is full.
For \(f\in C_c(\R)\) and \(n\in\Z\), direct convolution
gives
\begin{equation}\label{pt:corner-compression}
p(fv^n)p=g_{f,n}p,\qquad
g_{f,n}(t)=\sum_{k\in\Z}
\zeta(t-k)f(t-k)\zeta(t-k-n).
\end{equation}
The function \(g_{f,n}\) is continuous and one-periodic.
Conversely, for a continuous one-periodic \(g\), the
element \(gp\) belongs to \(E\), since all its Fourier
coefficients are compactly supported. Periodicity makes \(g\)
a central multiplier of \(E\), so \(gp\in pEp\), and the map
\(g\mapsto gp\) is a unital homomorphism
\(C(\R/\Z)\to pEp\). It is injective: its zeroth
coefficient is \(g\zeta^2\), and every orbit has a
point where \(\zeta\ne0\).
Equation~\eqref{pt:corner-compression} and density of
compactly supported Fourier sums prove surjectivity.
\end{proof}
\Needspace{3\baselineskip}
\begin{corollary}[Existence of point models]\label{pt:point-model}
There is a homomorphism \(\varphi:P\to Q_\infty\) such that
\[
\rho\circ\varphi=F(\rho|_Q)
\]
for every regularized limit \(\rho\).
\end{corollary}
\begin{proof}
Use \eqref{pt:crossed-map} and \Cref{pt:corner} to define
\(\varphi(a)=\Psi(p\otimes a)\).
For a regularized limit \(\rho\), the extended trace
\(\omega=\rho\circ\Psi\) restricts on
\(C_0(\R)\otimes P\) to Lebesgue measure tensored with
\(F(\sigma)\), where \(\sigma=\rho|_Q\).
Thus, for every \(a\in P_+\),
\[
\rho(\varphi(a))
=\omega(p\otimes a)
=\omega(\zeta^2\otimes a)
=F(\sigma)(a).
\]
The positive partition calculation applies equally when
this common value is infinite.
\end{proof}
\subsection{From a point model to an actual map}
The coordinate lifts of a point model need not converge.
Uniqueness will make their unitary orbits Cauchy on finite sets.
We can then conjugate a subsequence so that its successive
errors are summable and its limit lies in \(Q\).
\begin{proposition}[An actual homomorphism]\label{pt:actual-map}
There is a homomorphism \(\phi:P\to Q\) satisfying
\[
\tau\circ\phi=F(\tau)\qquad(\tau\in T(Q)),
\qquad \Cu(\phi)=G.
\]
It is injective, and its constant sequence is a point model.
\end{proposition}
\begin{proof}
Take c.p.c. lifts \(\varphi_n:P\to Q\) of the point
model in \Cref{pt:point-model}, using nuclearity and
\cite[Theorem~3.10]{ChoiEffrosLifting}.
For every finite \(\mathcal E\subseteq P\) and every
\(\eps>0\), there is \(N\) such that
\begin{equation}\label{pt:all-pairs}
\inf_{w\in\mathcal U(M(Q))}
\max_{a\in\mathcal E}
\|\Ad(w)(\varphi_m(a))-\varphi_n(a)\|<\eps
\qquad(m,n\geq N).
\end{equation}
If this failed, choose pairs \(m_k,n_k\to\infty\),
with both index sequences strictly increasing,
for which the displayed infimum is at least \(\eps\).
Every subsequence of a model is a model for the same
data: given coordinate traces $\tau_k$ on indices $n_k$ and a free
ultrafilter $\omega$ on $k$, set $\widetilde\tau_{n_k}=\tau_k$
and use the zero trace at the remaining coordinates. Define
$A\in\widetilde\omega$ if and only if
$\{k:n_k\in A\}\in\omega$. Then $\widetilde\omega$ is a free
ultrafilter concentrated on the chosen indices, and every cut
ultralimit agrees with the corresponding subsequence ultralimit. Thus \((\varphi_{m_k})\) and
\((\varphi_{n_k})\) define two point models.
\Cref{un:uniqueness} contradicts the choice of the pairs.
Let \(\mathcal E_k\) be increasing finite sets with dense
union in the unit ball of \(P\).
Use \eqref{pt:all-pairs} to choose an increasing
subsequence \(n_k\) and \(w_k\in\mathcal U(M(Q))\) with
\[
\max_{a\in\mathcal E_k}
\|\Ad(w_k)(\varphi_{n_{k+1}}(a))
-\varphi_{n_k}(a)\|<2^{-k}.
\]
Choose \(n_k\) beyond the threshold for the \(k\)-th
test, so that both indices in this inequality are
eligible. Starting with \(W_1=1\), set
\(W_{k+1}=W_k w_k\) and
\(\psi_k=\Ad(W_k)\circ\varphi_{n_k}\).
The sequence \(\psi_k(a)\) is Cauchy for every \(a\)
in the dense union, with summable successive errors,
and then for every \(a\in P\), by contractivity.
Its limit \(\phi:P\to Q\) is c.p.c. and multiplicative,
since the multiplicative defects of the original
coordinate lifts tend to zero. It is therefore a
homomorphism.
We check the invariant without assuming continuity of
unbounded trace evaluation under norm convergence.
Fix \(\tau\in T(Q)\), and let \(\rho_\tau\) be its
regularized constant-coordinate limit.
The maps \(\psi_k\) and the constant map \(\phi\)
give the same homomorphism into \(Q_\infty\).
For every \(a\in P_+\) and every cut tolerance,
coordinate unitary invariance gives
\[
\tau((\psi_k(a)-\eps)_+)
=\tau((\varphi_{n_k}(a)-\eps)_+).
\]
Regularization and the subsequence model identity
therefore imply
\[
\tau(\phi(a))
=\rho_\tau((\psi_k(a))_k)
=F(\tau)(a).
\]
Here \(\rho_\tau|_Q=\tau\) follows from
\Cref{seq:regularized}. Applying the equality to
positive powers gives
\[
d_\tau(\phi(a))=d_{F(\tau)}(a)
=d_\tau(G[a]).
\]
This calculation also applies to finite matrices. If
\(x=(x_{ij})\in M_k(P)_+\), the unnormalized matrix traces give
\begin{align*}
(\tau\otimes\Tr_k)(\phi^{(k)}(x))
&=\sum_i\tau(\phi(x_{ii}))
=\sum_i F(\tau)(x_{ii})\\
&=(F(\tau)\otimes\Tr_k)(x),
\end{align*}
including infinite values. Apply this identity to positive
powers and then use positive cuts of finite-matrix
approximations in \(P\otimes\K\). Pointwise trace
determination in \(\Cu(Q)\) gives \(\Cu(\phi)=G\).
Since \(G\) is injective, \(\phi(a)=0\) for positive
\(a\) implies \([a]=0\), hence \(a=0\);
thus \(\phi\) is injective.
Finally, for any regularized limit \(\rho\), put
\(\sigma=\rho|_Q\). Its value on the constant map is
\[
\rho\circ\phi=\sigma\circ\phi=F(\sigma).
\]
So the constant map is itself a point model.
\end{proof}
\subsection{The final intertwining}
\begin{theorem}[Trace-cone classification]\label{pt:intertwining}
There is an isomorphism \(\Phi:P\to Q\) such that
\[
\tau\circ\Phi=F(\tau)\quad(\tau\in T(Q)),
\qquad \Cu(\Phi)=G.
\]
Consequently \Cref{thm:main} holds.
\end{theorem}
\begin{proof}
Apply \Cref{pt:actual-map} to \(F\) and to \(F^{-1}\).
This gives homomorphisms
\(\phi:P\to Q\) and \(\psi:Q\to P\) with inverse
trace and Cuntz data.
Their compositions have identity trace data. The
constant maps \(\psi\phi\) and \(\id_P\) are therefore
point models for the same data, and similarly for
\(\phi\psi\) and \(\id_Q\).
By \Cref{un:uniqueness}, the compositions can be
conjugated arbitrarily close to the corresponding
identity on each finite set.
We give the intertwining construction to ensure that
the conclusion is an isomorphism in \(P,Q\).
Fix increasing finite sets \(\mathcal P_n,\mathcal Q_n\)
with dense unions in the unit balls, and put
\(\eps_n=2^{-n}\).
Start with \(f_1=\phi\), \(g_0=\psi\), and empty
\(\mathcal E_0,\mathcal F_0\).
Suppose \(f_n\) and \(g_{n-1}\) have been chosen.
Choose a finite set
\[
\mathcal E_n\supseteq
\mathcal E_{n-1}\cup\mathcal P_n
\cup g_{n-1}(\mathcal F_{n-1}).
\]
The composition \(g_{n-1}f_n\) still has identity
trace data, because every adjustment made below is
unitary conjugacy.
Choose \(v_n\in\mathcal U(M(P))\), and set
\(g_n=\Ad(v_n)\circ g_{n-1}\), so that
\begin{equation}\label{pt:left-intertwining}
\|g_nf_n(a)-a\|<\eps_n
\qquad(a\in\mathcal E_n).
\end{equation}
Next choose a finite set
\[
\mathcal F_n\supseteq
\mathcal F_{n-1}\cup\mathcal Q_n\cup f_n(\mathcal E_n).
\]
Choose \(z_n\in\mathcal U(M(Q))\), and set
\(f_{n+1}=\Ad(z_n)\circ f_n\), with
\begin{equation}\label{pt:right-intertwining}
\|f_{n+1}g_n(b)-b\|<\eps_n
\qquad(b\in\mathcal F_n).
\end{equation}
This second adjustment is possible by the same
identity-data argument for \(f_ng_n\).
The inclusions of the image tests make these sequences
Cauchy. If \(a\in\mathcal E_n\), contractivity and
\eqref{pt:left-intertwining}--\eqref{pt:right-intertwining}
give
\begin{align*}
\|f_{n+1}(a)-f_n(a)\|
&\leq\|f_{n+1}(a-g_nf_n(a))\|\\
&\quad+\|f_{n+1}g_n(f_n(a))-f_n(a)\|
<2\eps_n.
\end{align*}
If \(n\geq2\) and \(b\in\mathcal F_{n-1}\), then
\(g_{n-1}(b)\in\mathcal E_n\), and
\begin{align*}
\|g_n(b)-g_{n-1}(b)\|
&\leq\|g_n(b-f_ng_{n-1}(b))\|\\
&\quad+\|g_nf_n(g_{n-1}(b))-g_{n-1}(b)\|
<\eps_{n-1}+\eps_n.
\end{align*}
These summable bounds and density give point-norm
limits \(f_\infty:P\to Q\) and
\(g_\infty:Q\to P\), both homomorphisms.
Equations~\eqref{pt:left-intertwining} and
\eqref{pt:right-intertwining} imply
\[
g_\infty f_\infty=\id_P,\qquad
f_\infty g_\infty=\id_Q.
\]
For example, contractivity controls the moving input
\(f_n(a)\) by its limit \(f_\infty(a)\), after which
point-norm convergence of \(g_n\) applies to the fixed
limit input. Density gives these equalities on the
whole algebras. Thus \(f_\infty\) is the required isomorphism.
The isomorphism realizes the prescribed trace data as well. Each
\(f_n\) has Cuntz data \(G\), since the changes to \(\phi\) are
unitary conjugacies. For \(a\in P_+\) and \(\eps>0\), choose \(n\)
with \(\|f_n(a)-f_\infty(a)\|<\eps\). Positive cut comparison gives,
for every \(\tau\in T(Q)\),
\[
d_{F\tau}((a-\eps)_+)\leq d_\tau(f_\infty(a)),\qquad
d_\tau((f_\infty(a)-\eps)_+)\leq d_{F\tau}(a).
\]
Taking the supremum over cuts proves equality of ranks; spectral
integration gives \(\tau\circ f_\infty=F(\tau)\), including infinite
values. This argument uses no continuity of an unbounded weight on
an uncut norm-convergent sequence. The matrix argument in
\Cref{pt:actual-map} also gives \(\Cu(f_\infty)=G\).
Set \(\Phi=f_\infty\). Substituting
\(P=A\otimes\W\otimes\K\) and
\(Q=B\otimes\W\otimes\K\) proves \Cref{thm:main}.
\end{proof}
==== END The-trace-cone-classifies-Razak-Jacelon-stabilizations//sections/08-point-intertwining.tex ====
==== BEGIN The-trace-cone-classifies-Razak-Jacelon-stabilizations//references.bib ====
@article{ERS,
author = {Elliott, George A. and Robert, Leonel and Santiago, Luis},
title = {The cone of lower semicontinuous traces on a {$C^*$}-algebra},
journal = {American Journal of Mathematics},
volume = {133},
number = {4},
year = {2011},
pages = {969--1005},
url = {https://arxiv.org/abs/0805.3122v2}
}
@article{Robert,
author = {Robert, Leonel},
title = {The cone of functionals on the {Cuntz} semigroup},
journal = {Mathematica Scandinavica},
volume = {113},
number = {2},
year = {2013},
pages = {161--186},
doi = {10.7146/math.scand.a-15568},
url = {https://arxiv.org/abs/1102.1451v4}
}
@article{Jacelon,
author = {Jacelon, Bhishan},
title = {A simple, monotracial, stably projectionless {$C^*$}-algebra},
journal = {Journal of the London Mathematical Society},
series = {2},
volume = {87},
number = {2},
year = {2013},
pages = {365--383},
doi = {10.1112/jlms/jds049},
url = {https://arxiv.org/abs/1006.5397v3}
}
@article{Nawata,
author = {Nawata, Norio},
title = {A characterization of the {Razak--Jacelon} algebra},
journal = {Analysis \& PDE},
volume = {16},
number = {8},
year = {2023},
pages = {1799--1824},
doi = {10.2140/apde.2023.16.1799},
url = {https://arxiv.org/abs/2008.10235v3}
}
@article{Gabe,
author = {Gabe, James},
title = {Classification of {$\mathcal{O}_\infty$}-stable {$C^*$}-algebras},
journal = {Memoirs of the American Mathematical Society},
volume = {293},
number = {1461},
year = {2024},
url = {https://arxiv.org/abs/1910.06504v3}
}
@article{CGNN,
author = {Ciuperca, Alin and Giordano, Thierry and Ng, Ping Wong and Niu, Zhuang},
title = {Amenability and uniqueness},
journal = {Advances in Mathematics},
volume = {240},
year = {2013},
pages = {325--345},
doi = {10.1016/j.aim.2013.02.018},
url = {https://arxiv.org/abs/1207.6741v1}
}
@article{Voiculescu,
author = {Voiculescu, Dan-Virgil},
title = {A note on quasi-diagonal {$C^*$}-algebras and homotopy},
journal = {Duke Mathematical Journal},
volume = {62},
number = {2},
year = {1991},
pages = {267--271},
url = {https://doi.org/10.1215/S0012-7094-91-06211-3}
}
@misc{Shulman,
author = {Shulman, Tatiana},
title = {Sections and cones},
year = {2026},
howpublished = {arXiv:2507.22783v3},
note = {Version dated 11 May 2026},
url = {https://arxiv.org/abs/2507.22783v3}
}
@misc{STW,
author = {Schafhauser, Christopher and Tikuisis, Aaron and White, Stuart},
title = {Nuclear {$C^*$}-algebras: 99 problems},
year = {2026},
howpublished = {arXiv:2506.10902v2},
note = {Version dated 11 May 2026; to appear in M\"unster Journal of Mathematics},
url = {https://arxiv.org/abs/2506.10902v2}
}
@misc{ElliottNiu,
author = {Elliott, George A. and Niu, Zhuang},
title = {The classification of simple separable {$KK$}-contractible {$C^*$}-algebras with finite nuclear dimension},
year = {2016},
howpublished = {arXiv:1611.05159v1},
url = {https://arxiv.org/abs/1611.05159v1}
}
@article{EGLN,
author = {Elliott, George A. and Gong, Guihua and Lin, Huaxin and Niu, Zhuang},
title = {The classification of simple separable {$KK$}-contractible {$C^*$}-algebras with finite nuclear dimension},
journal = {Journal of Geometry and Physics},
volume = {158},
year = {2020},
pages = {103861},
doi = {10.1016/j.geomphys.2020.103861},
url = {https://arxiv.org/abs/1712.09463v5}
}
@article{DadarlatEilers,
author = {Dadarlat, Marius and Eilers, S{\o}ren},
title = {Asymptotic unitary equivalence in {$KK$}-theory},
journal = {{$K$}-Theory},
volume = {23},
number = {4},
year = {2001},
pages = {305--322},
url = {https://www.math.purdue.edu/~mdd/Publications/kk-asympt-with-eilers.pdf}
}
@article{Connes,
author = {Connes, Alain},
title = {Classification of injective factors. {Cases $II_1$, $II_\infty$, $III_\lambda$, $\lambda\ne1$}},
journal = {Annals of Mathematics},
series = {2},
volume = {104},
number = {1},
year = {1976},
pages = {73--115},
doi = {10.2307/1971057},
url = {https://annals.math.princeton.edu/1976/104-1/p04}
}
@article{HRS,
author = {Haagerup, Uffe and Rosenthal, Haskell P. and Sukochev, Fedor A.},
title = {Banach embedding properties of non-commutative {$L^p$}-spaces},
journal = {Memoirs of the American Mathematical Society},
volume = {163},
number = {776},
year = {2003},
pages = {1--68},
url = {https://arxiv.org/abs/math/0005150v1}
}
@article{VaesWouters,
author = {Vaes, Stefaan and Wouters, Lise},
title = {{Borel} fields and measured fields of {Polish} spaces, {Banach} spaces, von {Neumann} algebras and {$C^*$}-algebras},
journal = {Journal of the London Mathematical Society},
volume = {111},
number = {4},
year = {2025},
pages = {e70159},
doi = {10.1112/jlms.70159},
url = {https://arxiv.org/abs/2405.16603v2}
}
@article{ChoiEffros,
author = {Choi, Man-Duen and Effros, Edward G.},
title = {Nuclear {$C^*$}-algebras and the approximation property},
journal = {American Journal of Mathematics},
volume = {100},
number = {1},
year = {1978},
pages = {61--79},
doi = {10.2307/2373876},
url = {https://www.jstor.org/stable/2373876}
}
@article{ChoiEffrosLifting,
author = {Choi, Man-Duen and Effros, Edward G.},
title = {The completely positive lifting problem for {$C^*$}-algebras},
journal = {Annals of Mathematics},
series = {2},
volume = {104},
number = {3},
year = {1976},
pages = {585--609},
doi = {10.2307/1970968},
url = {https://annals.math.princeton.edu/1976/104-3/p08}
}
@article{Kasparov,
author = {Kasparov, Gennadi G.},
title = {Hilbert {$C^*$}-modules: theorems of {Stinespring} and {Voiculescu}},
journal = {Journal of Operator Theory},
volume = {4},
number = {1},
year = {1980},
pages = {133--150},
url = {https://jot.theta.ro/jot/archive/1980-004-001/1980-004-001-007.html}
}
@book{Lance,
author = {Lance, E. Christopher},
title = {Hilbert {$C^*$}-modules: A toolkit for operator algebraists},
series = {London Mathematical Society Lecture Note Series},
volume = {210},
publisher = {Cambridge University Press},
address = {Cambridge},
year = {1995},
url = {https://doi.org/10.1017/CBO9780511526206}
}
@article{DixmierPrime,
author = {Dixmier, Jacques},
title = {Sur les {$C^*$}-alg\`ebres},
journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
volume = {88},
year = {1960},
pages = {95--112},
doi = {10.24033/bsmf.1545},
url = {https://www.numdam.org/item/10.24033/bsmf.1545.pdf}
}
@article{GuptaJain,
author = {Gupta, Ved Prakash and Jain, Ranjana},
title = {On closed {Lie} ideals of certain tensor products of {$C^*$}-algebras},
journal = {Mathematische Nachrichten},
volume = {291},
number = {8--9},
year = {2018},
pages = {1297--1309},
doi = {10.1002/mana.201700009},
url = {https://arxiv.org/abs/1701.02218v2}
}
@article{KaplanskyDensity,
author = {Kaplansky, Irving},
title = {A theorem on rings of operators},
journal = {Pacific Journal of Mathematics},
volume = {1},
number = {2},
year = {1951},
pages = {227--232},
url = {https://msp.org/pjm/1951/1-2/pjm-v1-n2-p06-p.pdf}
}
@article{Haagerup,
author = {Haagerup, Uffe},
title = {Quasitraces on exact {$C^*$}-algebras are traces},
journal = {C. R. Math. Rep. Acad. Sci. Canada},
volume = {36},
number = {2--3},
year = {2014},
pages = {67--92},
url = {https://arxiv.org/abs/1403.7653v1}
}
@article{Razak,
author = {Razak, Shaloub},
title = {On the classification of simple stably projectionless {$C^*$}-algebras},
journal = {Canadian Journal of Mathematics},
volume = {54},
number = {1},
year = {2002},
pages = {138--224},
url = {https://doi.org/10.4153/CJM-2002-006-7}
}
@article{CastillejosEvington,
author = {Castillejos, Jorge and Evington, Samuel},
title = {Nuclear dimension of simple stably projectionless {$C^*$}-algebras},
journal = {Analysis \& PDE},
volume = {13},
number = {7},
year = {2020},
pages = {2205--2240},
doi = {10.2140/apde.2020.13.2205},
url = {https://arxiv.org/abs/1901.11441v4}
}
@article{RordamZ,
author = {R{\o}rdam, Mikael},
title = {The stable and the real rank of {$\mathcal Z$}-absorbing {$C^*$}-algebras},
journal = {International Journal of Mathematics},
volume = {15},
number = {10},
year = {2004},
pages = {1065--1084},
url = {https://arxiv.org/abs/math/0408020v2}
}
@article{GabeO2,
author = {Gabe, James},
title = {A new proof of {Kirchberg's} {$\mathcal O_2$}-stable classification},
journal = {Journal f{\"u}r die reine und angewandte Mathematik},
volume = {761},
year = {2020},
pages = {247--289},
doi = {10.1515/crelle-2018-0010},
url = {https://arxiv.org/abs/1706.03690v2}
}
@misc{SantiagoWStable,
author = {Santiago, Luis},
title = {{$\mathcal W$}-stable {$C^*$}-algebras},
year = {2012},
howpublished = {2012 West Coast Operator Algebra Seminar abstract},
note = {p.~4},
url = {https://pages.uoregon.edu/ncp/Conferences/WCOAS2012/Abstracts.pdf}
}
@article{ChoiEffrosInjectivity,
author = {Choi, Man-Duen and Effros, Edward G.},
title = {Nuclear {$C^*$}-algebras and injectivity: The general case},
journal = {Indiana University Mathematics Journal},
volume = {26},
number = {3},
year = {1977},
pages = {443--446},
url = {https://doi.org/10.1512/iumj.1977.26.26034}
}
@book{Blackadar,
author = {Blackadar, Bruce},
title = {Operator Algebras: Theory of {$C^*$}-Algebras and von Neumann Algebras},
series = {Encyclopaedia of Mathematical Sciences},
volume = {122},
publisher = {Springer},
address = {Berlin, Heidelberg},
year = {2006},
doi = {10.1007/3-540-28517-2},
url = {https://www.bruceblackadar.com/Mathematics/Cycr.pdf},
note = {Revised and corrected author version dated February 8, 2017}
}
@article{HanPaulsen,
author = {Han, Kyung Hoon and Paulsen, Vern I.},
title = {An approximation theorem for nuclear operator systems},
journal = {Journal of Functional Analysis},
volume = {261},
number = {4},
year = {2011},
pages = {999--1009},
doi = {10.1016/j.jfa.2011.04.009},
url = {https://arxiv.org/abs/1009.2541v4}
}
@misc{ConnesReport,
author = {Connes, Alain},
title = {On the classification of von {Neumann} algebras and their automorphisms},
year = {1976},
month = feb,
howpublished = {Institut des Hautes {\'E}tudes Scientifiques, Bures-sur-Yvette, report},
note = {{IHES} archive identifier P\_76\_132},
url = {https://repo-archives.ihes.fr/FONDS_IHES/I_Prepublications/CONNES/1976-1984/P_76_132/P_76_132_web.pdf}
}
@misc{OpenAINuclearDimension,
author = {{OpenAI}},
title = {{Nuclear dimension and Jiang--Su stability without elementary subquotients}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Nuclear-dimension-and-Jiang-Su-stability-without-elementary-subquotients-September-23-2026/paper.pdf}{OAI:Nuclear-dimension-and-Jiang-Su-stability-without-elementary-subquotients-September-23-2026}},
year = {2026}
}
@article{Schafhauser,
author = {Schafhauser, Christopher},
title = {A new proof of the {Tikuisis--White--Winter} theorem},
journal = {Journal f{\"u}r die reine und angewandte Mathematik},
volume = {759},
year = {2020},
pages = {291--304},
doi = {10.1515/crelle-2017-0056},
url = {https://arxiv.org/abs/1701.02180v3}
}
@article{BGSW,
author = {Bosa, Joan and Gabe, James and Sims, Aidan and White, Stuart},
title = {The nuclear dimension of {$\mathcal O_\infty$}-stable {$C^*$}-algebras},
journal = {Advances in Mathematics},
volume = {401},
year = {2022},
pages = {108250},
doi = {10.1016/j.aim.2022.108250},
url = {https://arxiv.org/abs/1906.02066v2}
}
@article{LinNg,
author = {Lin, Huaxin and Ng, Ping Wong},
title = {Extensions of {$C^*$}-Algebras by a Small Ideal},
journal = {International Mathematics Research Notices},
volume = {2023},
number = {12},
year = {2023},
pages = {10350--10438},
doi = {10.1093/imrn/rnac115},
url = {https://arxiv.org/abs/2006.00132v1}
}
==== END The-trace-cone-classifies-Razak-Jacelon-stabilizations//references.bib ====