64c3814
==== TABLE OF CONTENTS ====
Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings//circle.tex
Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings//introduction.tex
Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings//main.tex
Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings//near.tex
Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings//obstruction.tex
Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings//references.bib
===========================
==== BEGIN Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings//circle.tex ====
\section{A circle family with uniform Fourier bounds}\label{sec:circle}
We first construct representations in which a scalar circle parameter can
later be replaced by a unitary matrix. Two properties are needed: Fourier
coefficients with a summable first moment, uniformly over the group, and
a weak operator limit that retains the parameter.
Let $F$ be the free group on generators $s_j$, $j\geq 1$, with identity
$e$. Put $K=\ell^2(F)$, write $(\delta_x)_{x\in F}$ for its standard
orthonormal basis, and let $p_e$ be the projection onto
$\mathbb C\delta_e$. We use the left regular representation
$l(g)\delta_x=\delta_{gx}$. Throughout the construction,
\[
t=\frac12,\qquad r=(1-t^2)^{1/2},\qquad
\mathbb T=\{z\in\mathbb C:|z|=1\}.
\]
\begin{proposition}\label{prop:circle}
There are unitary representations $\pi_z:F\to\B(K)$, indexed by
$z\in\mathbb T$, and operators $A_k(g)\in\B(K)$, indexed by
$k\in\mathbb Z$ and $g\in F$, such that
\begin{align}
\pi_z(g)&=\sum_{k\in\mathbb Z}z^kA_k(g),
&\sup_{g\in F}\|A_k(g)\|&\leq c_k,
&c_k&=\mathbf 1_{\{k=0\}}+\frac{2t^{|k|}}{1-t},
\label{eq:fourier}\\
\pi_z(s_j)&\xrightarrow[j\to\infty]{\WOT}tzp_e.
\label{eq:circle-limit}
\end{align}
The series in \eqref{eq:fourier} converges absolutely in operator norm,
uniformly in $g$ and $z$, and
\begin{equation}\label{eq:moments}
S_0:=\sum_{k\in\mathbb Z}c_k<\infty,
\qquad S_1:=\sum_{k\in\mathbb Z}|k|c_k<\infty.
\end{equation}
\end{proposition}
\begin{proof}
The underlying representation is the real-parameter Pytlik--Szwarc
representation \cite[Section~2.3, Equation~(5)]{PS1986}; its cyclic
coefficient is the positive-definite function $g\mapsto t^{|g|}$
of Haagerup \cite[Lemma~1.2]{Haa1979}. We give the tree construction
before introducing the circle parameter.
Use the Cayley tree whose edges join $x$ to $xs_j^{\pm1}$, rooted at
$e$, and write $p(x)$ for the parent of $x\ne e$. Define
\begin{equation}\label{eq:tree-vectors}
\xi_e=\delta_e,\qquad
\xi_x=t\xi_{p(x)}+r\delta_x\quad(x\ne e).
\end{equation}
All these vectors have norm one. If $v$ is the last common vertex of the
paths from $e$ to $x$ and $y$, the two tails beyond $v$ are orthogonal
to each other and to $\xi_v$. Thus
\[
\langle\xi_x,\xi_y\rangle
=t^{d(x,v)+d(y,v)}=t^{d(x,y)},
\]
where $d$ is the tree distance. Their linear span contains every
$\delta_x$, by \eqref{eq:tree-vectors}. Left multiplication preserves
$d$, so the rule
\[
\pi(g)\xi_x=\xi_{gx}
\]
extends to a unitary on $K$. These unitaries form a representation.
The difference from the regular representation is confined to a single
finite path. To see this precisely, put $T_g=\pi(g)l(g)^*$.
For $y\ne g$, \eqref{eq:tree-vectors} gives
\begin{equation}\label{eq:root-change}
T_g\delta_y=
\frac{\xi_y-t\xi_{g p(g^{-1}y)}}{r},
\qquad T_g\delta_g=\xi_g.
\end{equation}
The vertex $g p(g^{-1}y)$ is the first step from $y$ toward $g$.
Let $e=h_0,h_1,\ldots,h_d=g$ be the path from $e$ to $g$.
For $y$ outside this path, the first steps toward $e$ and $g$
coincide; hence $T_g\delta_y=\delta_y$. On the path,
\begin{equation}\label{eq:path-action}
T_g\delta_{h_b}=r\xi_{h_b}-t\delta_{h_{b+1}}\quad(0\leq b<d),
\qquad T_g\delta_{h_d}=\xi_{h_d}.
\end{equation}
In particular, the path subspace and its orthogonal complement reduce
$T_g$. Expanding the vectors along the path gives
\[
\xi_{h_b}=t^b\delta_{h_0}
+r\sum_{a=1}^b t^{b-a}\delta_{h_a}.
\]
By \eqref{eq:path-action}, the subdiagonal entries are $-t$; every
other nonzero entry in row $a$, column $b$ is $t^{|a-b|}$ times
$1$, $r$, or $r^2$. Thus the matrix entries on this subspace satisfy
\begin{equation}\label{eq:path-decay}
|(T_g)_{a,b}|\leq t^{|a-b|}\qquad(0\leq a,b\leq d).
\end{equation}
This also includes $g=e$, when $T_e=I$.
Let $q:F\to\mathbb Z$ be the homomorphism with $q(s_j)=1$ for every
$j$, and define $D_z\delta_x=z^{-q(x)}\delta_x$. The desired
representation is
\begin{equation}\label{eq:circle-twist}
\pi_z(g)=z^{q(g)}D_z\pi(g)D_z^*
=(D_zT_gD_z^*)l(g).
\end{equation}
The first expression proves that it is a unitary representation. The
second follows from $D_zl(g)D_z^*=z^{-q(g)}l(g)$ and supplies its
Fourier bounds.
On the path subspace, the entry in row $a$, column $b$ acquires the
Laurent factor $z^{q(h_b)-q(h_a)}$. Each edge changes $q$ by $1$ or
$-1$, so an entry of exponent $k$ can occur only when $|a-b|\geq|k|$.
For each row or column there are at most two entries at any given
distance $m$. By \eqref{eq:path-decay}, the absolute row and column
sums of the coefficient of $z^k$ are therefore at most
\[
2\sum_{m\geq|k|}t^m=\frac{2t^{|k|}}{1-t}.
\]
Outside the path the operator is the identity, which contributes only
to the coefficient at $k=0$. The row and column sum bound for operator
norm, followed by multiplication by $l(g)$, proves
\eqref{eq:fourier}. For each fixed $g$ this is a finite Laurent
expansion; the common bounds $c_k$ give the asserted uniform absolute
convergence and \eqref{eq:moments}.
Finally, \eqref{eq:path-action} and \eqref{eq:circle-twist} give
\[
\pi_z(s_j)\delta_e=tz\delta_e+r\delta_{s_j}.
\]
For any fixed $x\ne e$, all sufficiently large $j$ satisfy
$s_jx\notin\{e,s_j\}$, and then
$\pi_z(s_j)\delta_x=\delta_{s_jx}$. These basis-vector calculations,
together with the uniform norm bound $\|\pi_z(s_j)\|=1$, prove
\eqref{eq:circle-limit}.
\end{proof}
==== END Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings//circle.tex ====
==== BEGIN Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings//introduction.tex ====
\section{Introduction}\label{sec:introduction}
Let $M$ and $N$ be von Neumann algebras acting on a complex Hilbert
space $H$, both with identity $I_H$. Their one-sided near-inclusion gap is
\begin{equation}\label{eq:gamma-definition}
\gamma(M,N)=\sup_{\substack{x\in M\\\|x\|\leq 1}}
\inf_{y\in N}\|x-y\|.
\end{equation}
The approximating element $y$ is not required to be contractive. A small
value of $\gamma(M,N)$ says that every contraction in $M$ is close to some
element of $N$; it does not require these choices of approximants to
preserve algebraic operations. If a unitary $u$ satisfies $uMu^*\subseteq N$,
then $\gamma(M,N)\leq 2\|u-I_H\|$. The converse question asks whether a
small near inclusion can always be obtained in this way.
The underlying perturbation program began with Kadison and Kastler
\cite{KK1972}. They measured the distance between operator algebras by
the Hausdorff distance between their unit balls in the operator norm,
now called the Kadison--Kastler distance, and proposed that sufficiently
close von Neumann algebras should be related by a unitary close to the
identity. Christensen, Johnson, and Raeburn--Taylor established the
amenable case in the 1970s
\cite{Chr1977,Joh1977,RT1977}. Christensen subsequently introduced
near inclusions and proved a spatial embedding theorem for an amenable
source algebra and an arbitrary target
\cite[Definition~2.1 and Theorem~4.3]{Chr1980}. In particular, sufficiently
small near inclusions in this setting admit implementing unitaries whose
distance from the identity is bounded linearly by the near-inclusion
tolerance.
We use the following formulation of the unrestricted one-sided question
posed by Cameron, Christensen, Sinclair, Smith, White, and Wiggins in
the paragraph preceding \cite[Theorem~2]{CCSSWW2012}:
\begin{quote}
For every $\varepsilon>0$, is there a $\delta>0$ such that, for every
complex Hilbert space $H$ and every pair of von Neumann algebras
$M,N\subseteq\mathcal B(H)$ with common identity $I_H$,
$\gamma(M,N)<\delta$ implies the existence of a unitary
$u\in\mathcal B(H)$ satisfying
\[
uMu^*\subseteq N
\qquad\text{and}\qquad
\|u-I_H\|<\varepsilon?
\]
\end{quote}
Here $\delta$ must be independent of the algebras and their
representations. We answer this question negatively.
\begin{theorem}\label{thm:main}
There exist $\varepsilon_0>0$, separable complex Hilbert spaces $H_n$,
and von Neumann algebras $M_n,N_n\subseteq\mathcal B(H_n)$ with common
identity $I_{H_n}$ such that
\[
\gamma(M_n,N_n)\longrightarrow 0,
\]
but, for all sufficiently large $n$, every unitary
$u\in\mathcal B(H_n)$ satisfying $uM_nu^*\subseteq N_n$ obeys
\[
\|u-I_{H_n}\|\geq\varepsilon_0.
\]
\end{theorem}
In these examples a unitary embedding of $M_n$ into $N_n$ does exist;
the obstruction concerns its distance from the identity. The theorem
therefore separates normwise approximation from small spatial
implementation even on separable Hilbert spaces. Its hypothesis is
one-sided. The construction supplies no reverse near inclusion, and
hence does not give a counterexample to the two-sided Kadison--Kastler
conjecture. The positive results for nonamenable factors in
\cite{CCSSWW2012}, which concern two-sided perturbations under additional
structural hypotheses, address a different assertion.
The proof transfers a classical finite-dimensional obstruction into the
near-inclusion problem. Voiculescu's clock and shift matrices
$U_n,V_n\in\mathcal B(\mathbb C^n)$ satisfy
$\|U_nV_n-V_nU_n\|\to0$, but cannot be approximated in norm by commuting
unitary pairs \cite{Voi1983}. Exel and Loring gave an elementary
winding-number proof of this phenomenon \cite{EL1989}; we use its trace-logarithm form
\cite[Lemma~3.1]{Exel1993}. We recall the short obstruction argument
in Section~\ref{sec:obstruction}.
To place these matrices in operator algebras, we use the free group
$\F$ on countably many generators. The starting representation
is the real-parameter Pytlik--Szwarc representation
\cite[\S2.3, Equation~(5)]{PS1986}, whose cyclic coefficient is Haagerup's
positive-definite function $g\mapsto t^{|g|}$, $0<t<1$, where $|g|$ is
reduced word length \cite[Lemma~1.2]{Haa1979}.
Section~\ref{sec:circle} gives a self-contained
tree realization at $t=1/2$ and adds a circle parameter $z$.
The norms of its Fourier coefficients are bounded, uniformly over the
group, by a sequence with summable first moment. Along successive
distinct free generators, the values of the representations converge
in the weak operator topology to $tz$ times the projection onto the
identity vector in $\ell^2(\F)$. Substituting $U_n$ and $V_n$ for the
circle parameter therefore produces two unitary representations of
$\F$ whose weak limits retain the original matrices.
Two further steps connect this representation family to the theorem.
In Section~\ref{sec:near}, a diagonal unitary built from the two
representations conjugates a copy of the group von Neumann algebra of
$\F\times\F$. The target $N_n$ is the commutant of an amplified right
regular representation of this product group. Reordering the Fourier
expansions gives an approximant in $N_n$ and an error bounded by a
constant times $\|U_nV_n-V_nU_n\|$. Crucially, this estimate holds on the
entire unit ball of the source algebra. In
Section~\ref{sec:obstruction}, a hypothetical sequence of spatial
embeddings tending to the identity places the conjugated left regular
representation in $N_n$, where it commutes with the right regular
representation. Their pointwise product is then a representation of
$\F\times\F$, so the von Neumann algebras generated by its two factors
commute. Weak operator
limits in these algebras recover operators close to copies of
$U_n$ and $V_n$ supported on projections. Perturbing those projections,
compressing to a common finite-dimensional corner, and taking polar
parts would yield
commuting unitary matrices close to $U_n,V_n$, contradicting the
winding-number obstruction.
==== END Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings//introduction.tex ====
==== BEGIN Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings//main.tex ====
\documentclass[11pt]{article}
\usepackage[T1]{fontenc}
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\usepackage[margin=1.1in]{geometry}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{microtype}
\usepackage{needspace}
\usepackage[hidelinks]{hyperref}
\input{glyphtounicode}
\input{glyphtounicode-cmex}
\pdfgentounicode=1
\listfiles
\hypersetup{
pdftitle={Near Inclusions of von Neumann Algebras Without Small Spatial Embeddings},
pdfauthor={OpenAI},
pdfsubject={One-sided near inclusions of von Neumann algebras},
pdfkeywords={von Neumann algebras, near inclusion, almost commuting matrices}
}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\B}{\mathcal B}
\newcommand{\F}{F}
\newcommand{\G}{G}
\newcommand{\WOT}{\mathrm{WOT}}
\newcommand{\id}{\mathrm{id}}
\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\dist}{dist}
\DeclareMathOperator{\spec}{spec}
\numberwithin{equation}{section}
\title{Near Inclusions of von Neumann Algebras\\Without Small Spatial Embeddings}
\author{OpenAI}
\date{October 5, 2026}
\begin{document}
\maketitle
\begin{abstract}
We give a negative answer to the unrestricted small-spatial-embedding
problem for one-sided near inclusions of von Neumann algebras. On
separable complex Hilbert spaces, we construct pairs of unital von
Neumann algebras with common identity whose one-sided gaps tend to zero.
Spatial embeddings of the source into the target exist, but every
implementing unitary stays a fixed positive distance from the identity.
The obstruction therefore concerns small implementing unitaries, rather
than the existence of spatial embeddings.
\end{abstract}
\input{sections/introduction}
\input{sections/circle}
\input{sections/near}
\input{sections/obstruction}
\bibliographystyle{amsplain}
\bibliography{references}
\end{document}
==== END Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings//main.tex ====
==== BEGIN Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings//near.tex ====
\section{From a unitary pair to a near inclusion}\label{sec:near}
We now use Proposition~\ref{prop:circle} to encode two unitaries
$U,V$ on a nonzero finite-dimensional Hilbert space $D$. The resulting
von Neumann algebras will have one-sided gap bounded by a constant
times $\|UV-VU\|$.
\subsection{Matrix substitution and the algebras}
On $L=K\otimes K\otimes D$, define
\begin{equation}\label{eq:matrix-reps}
a(g)=\sum_k A_k(g)\otimes I_K\otimes U^k,
\qquad
b(g)=\sum_k I_K\otimes A_k(g)\otimes V^k
\quad(g\in F).
\end{equation}
The series converge absolutely in norm. A spectral decomposition
$U=\sum_z zP_z$ rewrites the first expression as
$\sum_z\pi_z(g)\otimes I_K\otimes P_z$, so $a$ is a unitary
representation. Diagonalizing $V$ gives the same conclusion for $b$.
There is no requirement that their ranges commute. The weak limits
from Proposition~\ref{prop:circle} become
\begin{equation}\label{eq:matrix-limits}
a(s_j)\xrightarrow{\WOT}t p_e\otimes I_K\otimes U,
\qquad
b(s_j)\xrightarrow{\WOT}t I_K\otimes p_e\otimes V.
\end{equation}
Only finite spectral sums are involved, so these limits follow
directly from \eqref{eq:circle-limit}.
Put $G=F\times F$, $S=\ell^2(G)$, and $H=S\otimes L$.
Our conventions for the commuting left and right regular
representations on $S$ are
\[
\lambda(g)\delta_x=\delta_{gx},\qquad
\rho(g)\delta_x=\delta_{xg^{-1}}.
\]
For $x=(x_1,x_2)\in G$, put $f(x)=a(x_1)b(x_2)$ and define the
diagonal unitary $W$ by
\[
W(\delta_x\otimes v)=\delta_x\otimes f(x)v.
\]
Although $a$ and $b$ are representations, $f$ need not be one.
Define
\begin{equation}\label{eq:algebras}
M=W\bigl(\lambda(G)''\otimes I_L\bigr)W^*,
\qquad N=\bigl(\rho(G)\otimes I_L\bigr)'.
\end{equation}
Primes denote commutants. Here $\lambda(G)''\otimes I_L$ means
the algebra of all $X\otimes I_L$ with $X\in\lambda(G)''$;
it is weak operator closed, as is seen by taking matrix blocks
relative to an orthonormal basis of $L$. Thus $M,N$ are von Neumann
algebras with common identity $I_H$. Notice also that
$W^*MW\subset N$, so these examples always admit a spatial embedding.
The exactly commuting case explains the choice of $W$. If $U$ and $V$
commute, then so do the ranges of $a$ and $b$, and $f$ is a
representation of $G$. In that case
\[
W(\lambda(g)\otimes I_L)W^*=\lambda(g)\otimes f(g)\in N
\quad(g\in G),
\]
so $M\subset N$. When $U$ and $V$ only nearly commute, we must
control the error on the entire unit ball of $M$; estimates for the
group elements alone would not suffice.
\subsection{A norm estimate on the whole algebra}
\begin{proposition}\label{prop:near}
For the algebras in \eqref{eq:algebras},
\begin{equation}\label{eq:near-bound}
\gamma(M,N)\leq C\|UV-VU\|,
\qquad
C:=\sum_{k,m,p,h\in\mathbb Z}
c_kc_mc_pc_h|m-p|\,|h|<\infty.
\end{equation}
Moreover, for every $g\in G$,
\begin{equation}\label{eq:group-bound}
\bigl\|W(\lambda(g)\otimes I_L)W^*
-\lambda(g)\otimes f(g)\bigr\|
\leq C\|UV-VU\|.
\end{equation}
The constant is independent of $D,U,V$.
\end{proposition}
\begin{proof}
Write $\nu=\|UV-VU\|$. Fix $X\in\lambda(G)''$ and denote its
scalar matrix entries by $X_{x,y}$, for $x,y\in G$. Since $X$
commutes with $\rho(G)$, these entries are invariant under common
right translation:
\begin{equation}\label{eq:right-invariance}
X_{xq,yq}=X_{x,y}\quad(x,y,q\in G).
\end{equation}
The operator $Z=W(X\otimes I_L)W^*$ has $\B(L)$-valued blocks
\[
Z_{x,y}=X_{x,y}a(x_1)b(x_2)b(y_2)^*a(y_1)^*.
\]
We seek an approximant $Y$ with the reordered blocks
\begin{equation}\label{eq:reordered-blocks}
Y_{x,y}=X_{x,y}a(x_1)a(y_1)^*b(x_2)b(y_2)^*
=X_{x,y}f(xy^{-1}).
\end{equation}
The last equality uses the representation laws for $a$ and $b$
separately. Since $(xq)(yq)^{-1}=xy^{-1}$, these blocks are
right-translation invariant by \eqref{eq:right-invariance}.
Thus, once boundedness is established,
$Y$ commutes with $\rho(G)\otimes I_L$ and belongs to $N$.
The Fourier bounds establish boundedness and control the reordering
in operator norm. On $S\otimes K\otimes K$, let $D_{k,m}$ be
the diagonal operator whose block at $x\in G$ is
$A_k(x_1)\otimes A_m(x_2)$. Then $\|D_{k,m}\|\leq c_kc_m$.
Set
\[
B_{k,m,p,h}(X)=
D_{k,m}(X\otimes I_K\otimes I_K)D_{h,p}^*.
\]
The $A_k$ coefficients of $a$ and $b$ act on separate copies of $K$;
only their matrix factors $U^k,V^m$ act on the same space. In the
decomposition $H=(S\otimes K\otimes K)\otimes D$, expansion of the
four factors therefore gives
\begin{align}
Z&=\sum_{k,m,p,h} B_{k,m,p,h}(X)
\otimes U^kV^{m-p}U^{-h},
\label{eq:z-expansion}\\
Y&=\sum_{k,m,p,h} B_{k,m,p,h}(X)
\otimes U^{k-h}V^{m-p}.
\label{eq:y-expansion}
\end{align}
Both sums converge absolutely because
\[
\|B_{k,m,p,h}(X)\|
\leq c_kc_mc_pc_h\|X\|,
\qquad \sum_{k,m,p,h}c_kc_mc_pc_h=S_0^4<\infty.
\]
Their blocks are precisely the expressions above, which also
justifies the definition of $Y$.
For integers $a,b$, telescoping gives
\begin{equation}\label{eq:power-commutator}
\|U^aV^b-V^bU^a\|\leq |a|\,|b|\nu.
\end{equation}
For positive powers, expand a commutator one factor at a time;
negative powers follow from
$[U^{-1},V]=-U^{-1}[U,V]U^{-1}$ and its analogue for $V^{-1}$,
where $[A,B]=AB-BA$. Consequently the two matrix factors in
\eqref{eq:z-expansion} and \eqref{eq:y-expansion} differ by at most
$|m-p|\,|h|\nu$. Hence
\begin{equation}\label{eq:reordering-estimate}
\|Z-Y\|\leq C\nu\|X\|,
\qquad C\leq 2S_0^2S_1^2<\infty.
\end{equation}
Since $Y\in N$ and $\|Z\|=\|X\|$, this proves
\eqref{eq:near-bound}; no bound of $1$ on $\|Y\|$ is needed.
If $X=\lambda(g)$, its nonzero entries have $x=gy$. The
representation laws for $a$ and $b$ turn
\eqref{eq:reordered-blocks} into
$Y=\lambda(g)\otimes a(g_1)b(g_2)=\lambda(g)\otimes f(g)$.
Thus \eqref{eq:group-bound} is the same estimate for this choice of $X$.
\end{proof}
==== END Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings//near.tex ====
==== BEGIN Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings//obstruction.tex ====
\section{Recovering the matrix obstruction}\label{sec:obstruction}
The preceding construction turns a small commutator into a small
one-sided gap. We now show that a spatial embedding close to the
identity would turn the original matrix pair into a nearby commuting
pair. Choosing the clock and shift matrices will make this impossible.
\subsection{The winding obstruction}
The matrices below are Voiculescu's almost commuting unitary pair
\cite{Voi1983}. We include the logarithmic winding argument, in the
form developed in \cite{EL1989,Exel1993}.
\begin{lemma}\label{lem:clock}
For $n\geq3$, let $U_n,V_n$ be the unitaries on $\mathbb C^n$ given by
\begin{equation}\label{eq:clock-shift}
U_ne_j=\omega_n^je_j,\qquad
V_ne_j=e_{j+1\bmod n},\qquad
\omega_n=e^{2\pi i/n}\quad(0\leq j<n).
\end{equation}
Then $\nu_n:=\|U_nV_n-V_nU_n\|=|\omega_n-1|\to0$.
There is no infinite subsequence along which commuting unitaries
$\widehat U_n,\widehat V_n$ on $\mathbb C^n$ satisfy
\[
\|\widehat U_n-U_n\|+\|\widehat V_n-V_n\|\longrightarrow0.
\]
\end{lemma}
\begin{proof}
The identity $U_nV_n=\omega_nV_nU_n$ proves the first assertion.
For any unitary matrices $U,V$ whose commutator $UVU^*V^*$ has no
eigenvalue $-1$, define
\[
\kappa(U,V)=\frac{1}{2\pi i}\Tr\log(UVU^*V^*),
\]
using the principal logarithm and the unnormalized trace.
The determinant identity
\[
\exp\bigl(\Tr\log(UVU^*V^*)\bigr)
=\det(UVU^*V^*)=1
\]
shows that $\kappa(U,V)$ is an integer. It is continuous, and hence
constant, along any path on which the logarithm remains defined.
For the pair in \eqref{eq:clock-shift}, its value is $1$, since
$U_nV_nU_n^*V_n^*=\omega_n I$ and $0<2\pi/n<\pi$.
For a commuting pair its value is $0$.
Suppose the asserted approximants existed along a subsequence.
For all sufficiently large $n$ on that subsequence, write
$U_n^*\widehat U_n=e^{iA_n}$ and
$V_n^*\widehat V_n=e^{iB_n}$, with $A_n,B_n$ self-adjoint and
spectra in $(-\pi,\pi)$. The paths
\[
U_n(s)=U_ne^{isA_n},\qquad
V_n(s)=V_ne^{isB_n}\qquad(0\leq s\leq1)
\]
stay within the respective endpoint distances from $U_n,V_n$.
Indeed, for $|\theta|<\pi$ and $0\leq s\leq1$,
$|e^{is\theta}-1|\leq|e^{i\theta}-1|$.
A four-factor telescoping estimate gives, uniformly in $s$,
\[
\|U_n(s)V_n(s)U_n(s)^*V_n(s)^*-I\|
\leq \nu_n+2\|\widehat U_n-U_n\|
+2\|\widehat V_n-V_n\|\longrightarrow0.
\]
The principal logarithm is therefore defined along the whole path
for all sufficiently large $n$. Its integer changes from $1$ to $0$,
a contradiction.
\end{proof}
\subsection{A finite-dimensional corner}
Compression of commuting operators to a prescribed subspace need not
preserve commutation. The next lemma avoids this difficulty by
replacing each support projection by a nearby projection in its
corresponding algebra and then using their common range.
\begin{lemma}\label{lem:corner}
For each $n$, let $E_{1n},E_{2n}$ be commuting projections on a complex
Hilbert space $H_n$, with $D_n=E_nH_n$ nonzero and finite-dimensional,
where $E_n=E_{1n}E_{2n}$. Let $\widetilde R_{1n},\widetilde R_{2n}$
be unitaries commuting with both projections, and write
$R_{in}=\widetilde R_{in}|_{D_n}$.
Suppose $\mathcal A_{1n},\mathcal A_{2n}\subset\B(H_n)$ are
commuting unital von Neumann algebras and there are
$T_{in}\in\mathcal A_{in}$ such that
\begin{equation}\label{eq:corner-hypothesis}
\|T_{in}-E_{in}\widetilde R_{in}\|\longrightarrow0
\quad(i=1,2).
\end{equation}
Then, for all sufficiently large $n$, there are commuting unitaries
$\widehat R_{1n},\widehat R_{2n}$ on $D_n$ with
\[
\|\widehat R_{in}-R_{in}\|\longrightarrow0\quad(i=1,2).
\]
\end{lemma}
\begin{proof}
We first replace the intended projections by nearby projections in
the commuting algebras. Set
$\delta_{in}=\|T_{in}-E_{in}\widetilde R_{in}\|$.
Since $(E_{in}\widetilde R_{in})^*(E_{in}\widetilde R_{in})=E_{in}$,
\[
d_{in}:=\|T_{in}^*T_{in}-E_{in}\|
\leq \delta_{in}(2+\delta_{in})\longrightarrow0.
\]
For sufficiently large $n$, take the spectral projection
\[
Q_{in}=\mathbf1_{(1/2,\infty)}(T_{in}^*T_{in})\in\mathcal A_{in}.
\]
The spectrum of $T_{in}^*T_{in}$ lies in
$[0,d_{in}]\cup[1-d_{in},1+d_{in}]$. This follows from the
resolvent bound for a self-adjoint operator within $d_{in}$ of a
projection. Functional calculus therefore gives
\[
\|Q_{in}-T_{in}^*T_{in}\|\leq d_{in},
\qquad \|Q_{in}-E_{in}\|\leq2d_{in}\longrightarrow0.
\]
Since the two algebras commute, $P_n=Q_{1n}Q_{2n}$ is a projection,
and
\[
\|P_n-E_n\|
\leq\|Q_{1n}-E_{1n}\|+\|Q_{2n}-E_{2n}\|\longrightarrow0.
\]
We identify this perturbed corner with $D_n$ by a small unitary.
The operator
\[
S_n=P_nE_n+(I-P_n)(I-E_n)
=I+(P_n-E_n)(2E_n-I)
\]
tends to the identity in norm and satisfies $S_nE_n=P_nS_n$.
It is invertible for large $n$. Its absolute value commutes with
$E_n$, so its unitary polar part
$J_n=S_n(S_n^*S_n)^{-1/2}$ satisfies
\begin{equation}\label{eq:corner-identification}
J_nE_nJ_n^*=P_n,\qquad \|J_n-I\|\longrightarrow0.
\end{equation}
In particular, $P_nH_n$ has exactly the dimension of $D_n$.
The compressions $P_nT_{1n}P_n$ and $P_nT_{2n}P_n$ commute,
including across adjoints. This needs a check because $Q_{in}$ need
not commute with $T_{in}$. Suppress $n$ and put
$A=Q_1T_1Q_1\in\mathcal A_1$ and
$B=Q_2T_2Q_2\in\mathcal A_2$. The two compressions are $AQ_2$
and $Q_1B$. Since $AQ_1=Q_1A=A$, $BQ_2=Q_2B=B$, and every
element of $\mathcal A_1$ commutes with every element of
$\mathcal A_2$, both orders of their product equal $AB$.
Replacing $A$ or $B$ by its adjoint proves the other commutations.
Transfer these compressions to $D_n$ using \eqref{eq:corner-identification}:
\[
C_{in}=\left.E_nJ_n^*T_{in}J_nE_n\right|_{D_n}.
\]
The pair still commutes across adjoints. Moreover,
\[
\|C_{in}-R_{in}\|
\leq\delta_{in}+2\|J_n-I\|\longrightarrow0,
\]
by comparing $T_{in}$ with $E_{in}\widetilde R_{in}$ before
conjugation. Thus $C_{in}$ is invertible for large $n$ and
$\|C_{in}^*C_{in}-I_{D_n}\|\to0$. Its unitary polar part
\[
\widehat R_{in}=C_{in}(C_{in}^*C_{in})^{-1/2}
\]
remains close to $R_{in}$. The unital $C^*$-algebras generated by
$C_{1n}$ and $C_{2n}$ commute; continuous functional calculus puts
each polar part in its respective algebra. Hence the two polar
unitaries commute, as required.
\end{proof}
\subsection{Proof of the main theorem}
Apply the construction of Section~\ref{sec:near} to
$D=\mathbb C^n$ and the clock--shift pair of Lemma~\ref{lem:clock}.
Use subscripts $n$ for the resulting objects $L_n,H_n,a_n,b_n,f_n,W_n,
M_n,N_n$. All these Hilbert spaces are separable. By
Proposition~\ref{prop:near},
\begin{equation}\label{eq:vanishing-gap}
\gamma(M_n,N_n)\leq C\nu_n\longrightarrow0.
\end{equation}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
We show that no infinite subsequence admits unitaries $u_n$ with
\begin{equation}\label{eq:assumed-embedding}
u_nM_nu_n^*\subset N_n,\qquad\|u_n-I\|\longrightarrow0.
\end{equation}
Suppose otherwise and work on that subsequence. Define, for $g\in G$,
\[
Z_n(g)=W_n(\lambda(g)\otimes I_{L_n})W_n^*,\qquad
\Sigma_n(g)=u_nZ_n(g)u_n^*(\rho(g)\otimes I_{L_n}).
\]
The first factor in the expression for $\Sigma_n(g)$ lies in $N_n$,
by \eqref{eq:assumed-embedding}. Thus the two representations being
multiplied have commuting ranges, and $\Sigma_n$ is a unitary
representation of $G$. Proposition~\ref{prop:near} gives the uniform
estimate
\begin{equation}\label{eq:sigma-bound}
\sup_{g\in G}
\bigl\|\Sigma_n(g)-\lambda(g)\rho(g)\otimes f_n(g)\bigr\|
\leq\beta_n,
\qquad \beta_n=C\nu_n+2\|u_n-I\|\longrightarrow0.
\end{equation}
Let $\mathcal A_{1n}$ and $\mathcal A_{2n}$ be the von Neumann
algebras generated by the restrictions of $\Sigma_n$ to $F\times\{e\}$
and $\{e\}\times F$, respectively. These algebras commute.
We will apply Lemma~\ref{lem:corner} by taking weak limits of the
generators in each factor. Under $S=K\otimes K$, write
\[
H_n=K\otimes K\otimes K\otimes K\otimes\mathbb C^n.
\]
Let $E_{1n}$ use $p_e$ in the first and third slots, and let $E_{2n}$
use $p_e$ in the second and fourth slots, with identities in all
other slots. These projections commute, and
$E_n=E_{1n}E_{2n}$ has range naturally identified with $\mathbb C^n$.
Let $\widetilde R_{1n}$ and $\widetilde R_{2n}$ act as $U_n,V_n$,
respectively, on the final slot and as the identity elsewhere.
Set $g_{1j}=(s_j,e)$ and $g_{2j}=(e,s_j)$. On the $i$-th factor
of $S$, the operator $\lambda(g_{ij})\rho(g_{ij})$ sends
$\delta_x$ to $\delta_{s_jxs_j^{-1}}$. For each fixed nonidentity
word $x$, this conjugate leaves every finite subset of $F$ as
$j\to\infty$: for large $j$, the letter $s_j$ occurs neither in
$x$ nor in the finitely many words under consideration. Hence the
conjugation operators converge weakly to $p_e$ in that slot.
Combining this observation with \eqref{eq:matrix-limits} gives,
for each fixed $n$ and $i$,
\begin{equation}\label{eq:joint-limit}
\lambda(g_{ij})\rho(g_{ij})\otimes f_n(g_{ij})
\xrightarrow[j\to\infty]{\WOT}tE_{in}\widetilde R_{in}.
\end{equation}
Here tensor products of bounded weakly convergent sequences on
separate slots converge weakly, as follows first on elementary
tensors and then by density. The finite spectral decompositions
used in \eqref{eq:matrix-limits} justify the matrix-valued factor.
For each $i,n$, choose a weak operator cluster point $B_{in}$ of
$\Sigma_n(g_{ij})$ as $j\to\infty$. Such a point exists by weak
operator compactness of the closed unit ball of $\B(H_n)$, and
belongs to $\mathcal A_{in}$. Norm balls are weak operator closed,
so \eqref{eq:sigma-bound} and \eqref{eq:joint-limit} imply
\[
\|B_{in}-tE_{in}\widetilde R_{in}\|\leq\beta_n.
\]
Thus $T_{in}=t^{-1}B_{in}\in\mathcal A_{in}$ satisfies
\eqref{eq:corner-hypothesis}. No product of weak limits is used:
commutation comes from membership in the two commuting algebras.
Lemma~\ref{lem:corner} now produces commuting unitary matrices on
$\mathbb C^n$ approaching $U_n,V_n$ in norm. This contradicts
Lemma~\ref{lem:clock}, proving that \eqref{eq:assumed-embedding}
cannot hold on any infinite subsequence.
It follows that some $\epsilon_0>0$ bounds
$\|u-I_{H_n}\|$ from below for every implementing unitary and all
sufficiently large $n$. Otherwise one could choose increasing
$n_j$ and implementing unitaries with distance less than $1/j$,
contrary to what we have just proved. Together with
\eqref{eq:vanishing-gap}, this is Theorem~\ref{thm:main}.
\end{proof}
==== END Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings//obstruction.tex ====
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