==== 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 b0$, 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 $g\mapsto t^{|g|}$, $00$ 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 ====
==== BEGIN Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings//references.bib ====
@article{KK1972,
author = {Kadison, Richard V. and Kastler, Daniel},
title = {Perturbations of von {Neumann} algebras. {I}. {Stability} of type},
journal = {American Journal of Mathematics},
volume = {94},
number = {1},
year = {1972},
pages = {38--54},
doi = {10.2307/2373592}
}
@article{Chr1977,
author = {Christensen, Erik},
title = {Perturbation of operator algebras},
journal = {Inventiones Mathematicae},
volume = {43},
number = {1},
year = {1977},
pages = {1--13},
doi = {10.1007/BF01390201}
}
@article{Joh1977,
author = {Johnson, Barry E.},
title = {Perturbations of {Banach} algebras},
journal = {Proceedings of the London Mathematical Society},
series = {3},
volume = {34},
number = {3},
year = {1977},
pages = {439--458},
doi = {10.1112/plms/s3-34.3.439}
}
@article{RT1977,
author = {Raeburn, Iain and Taylor, Joseph L.},
title = {Hochschild cohomology and perturbations of {Banach} algebras},
journal = {Journal of Functional Analysis},
volume = {25},
number = {3},
year = {1977},
pages = {258--266},
doi = {10.1016/0022-1236(77)90072-6}
}
@article{Chr1980,
author = {Christensen, Erik},
title = {Near inclusions of {$C^*$}-algebras},
journal = {Acta Mathematica},
volume = {144},
year = {1980},
pages = {249--265},
doi = {10.1007/BF02392125}
}
@article{CCSSWW2012,
author = {Cameron, Jan and Christensen, Erik and Sinclair, Allan M. and
Smith, Roger R. and White, Stuart A. and Wiggins, Alan D.},
title = {Type {$\mathrm{II}_1$} factors satisfying the spatial isomorphism conjecture},
journal = {Proceedings of the National Academy of Sciences of the United States of America},
volume = {109},
number = {50},
year = {2012},
pages = {20338--20343},
doi = {10.1073/pnas.1217792109}
}
@article{Haa1979,
author = {Haagerup, Uffe},
title = {An example of a non nuclear {$C^*$}-algebra, which has the metric approximation property},
journal = {Inventiones Mathematicae},
volume = {50},
number = {3},
year = {1979},
pages = {279--293},
doi = {10.1007/BF01410082}
}
@article{PS1986,
author = {Pytlik, Tadeusz and Szwarc, Ryszard},
title = {An analytic family of uniformly bounded representations of free groups},
journal = {Acta Mathematica},
volume = {157},
number = {3--4},
year = {1986},
pages = {287--309},
doi = {10.1007/BF02392596}
}
@article{Voi1983,
author = {Voiculescu, Dan},
title = {Asymptotically commuting finite rank unitary operators without commuting approximants},
journal = {Acta Scientiarum Mathematicarum (Szeged)},
volume = {45},
year = {1983},
pages = {429--431}
}
@article{EL1989,
author = {Exel, Ruy and Loring, Terry A.},
title = {Almost commuting unitary matrices},
journal = {Proceedings of the American Mathematical Society},
volume = {106},
number = {4},
year = {1989},
pages = {913--915},
doi = {10.1090/S0002-9939-1989-0975641-9}
}
@article{Exel1993,
author = {Exel, Ruy},
title = {The soft torus and applications to almost commuting matrices},
journal = {Pacific Journal of Mathematics},
volume = {160},
number = {2},
year = {1993},
pages = {207--217},
doi = {10.2140/pjm.1993.160.207}
}
==== END Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings//references.bib ====