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\title[On the isomorphism class of $q$-Gaussian W$^\ast$-algebras for infinite variables]{On the isomorphism class of $q$-Gaussian W$^\ast$-algebras for infinite variables}

\author{\firstname{Martijn} \lastname{Caspers}}
\address{TU Delft, EWI/DIAM, P.O.Box 5031, 2600 GA Delft, The Netherlands}
\email{M.P.T.Caspers@tudelft.nl}

\thanks{Supported by the NWO Vidi grant VI.Vidi.192.018 ``Non-commutative harmonic analysis and rigidity of operator algebras''}
\CDRGrant{NWO Vidi grant VI.Vidi.192.018}

\subjclass{46L35, 46L06}
\keywords{$q$-Gaussian von Neumann algebras, Akemann--Ostrand property}

\begin{abstract}
Let $M_q(H_{\mathbb{R}})$ be the $q$-Gaussian von Neumann algebra associated with a separable infinite dimensional real Hilbert space $H_{\mathbb{R}}$ where $-1 < q < 1$. We show that $M_q(H_{\mathbb{R}}) \not \simeq M_0(H_{\mathbb{R}})$ for $-1 < q \not = 0 < 1$. The C$^\ast$-algebraic counterpart of this result was obtained recently in~\cite{BCKW}. Using ideas of Ozawa we show that this non-isomorphism result also holds on the level of von Neumann algebras.
\end{abstract}

\begin{document}
\maketitle

\section{Introduction}

Von Neumann algebras of $q$-Gaussian variables originate from the work of Bo\.{z}ejko and Speicher~\cite{BS} (see also~\cite{BKS}). To a real Hilbert space $\RH$ and a parameter $-1 < q < 1$ it associates a von Neumann algebra $M_q(\RH)$. At parameter $q=0$ this assignment $\RH \mapsto M_q(\RH)$ is known as Voiculescu's free Gaussian functor. The dependence of $q$ of these von Neumann algebras has been an intriguing and very difficult problem.~A breakthrough result in this direction was obtained by Guionnet--Shlyakhtenko~\cite{GS} who showed that for finite dimensional $\RH$ for a range of $q$ close to 0 all von Neumann algebras $M_q(\RH)$ are isomorphic. The range for which isomorphism is known decreases as the dimension $\RH$ becomes larger. The Guionnet--Shlyakhtenko approach is based on free analogues of (optimal) transport techniques. Their result also relies on existence and power series estimates of conjugate variables obtained by Dabrowski~\cite{Dabrowski}. In fact the free transport techniques provide even an isomorphism result of underlying $q$-Gaussian C$^\ast$-algebras.

In case $\RH$ is infinite dimensional the isomorphism question of $q$-Gaussian algebras was addressed by Nelson and Zeng~\cite{NelsonZeng}. They showed that for \emph{mixed} $q$-Gaussians for which the array $(q_{ij})_{ij}$ of commutation coefficients decays fast enough to 0 one obtains isomorphism of mixed $q$-Gaussian C$^\ast$- and von Neumann algebras. However, the isomorphism question for the original (non-mixed) $q$-Gaussians remained open, see Questions 1.1 and 1.2 of~\cite{NelsonZeng}. In~\cite{BCKW} we showed that on the level of C$^\ast$-algebras there exists a non-isomorphsim result. In the current note we improve on this result: we show that for an infinite dimensional separable real Hilbert space $\RH$ and $-1 < q <1, q \not = 0$ we have $M_q(\RH) \not \simeq M_0(\RH)$. This then fully answers Questions 1.1 and 1.2 of~\cite{NelsonZeng} and provides a stark contrast to the results of Guionnet--Shlyakhtenko for finite dimensional $\RH$.

The distinguishing property of $M_q(\RH)$ and $M_0(\RH)$ is a variation of the Akemann--Ostrand property that was suggested in a note by Ozawa~\cite{OzawaNote} (see also~\cite{PetersonEtAl}) and which we shall call W$^\ast$AO. We formally define it in Definition~\ref{Eqn=WAO}. The most important novelty is that we quotient $\cB(L^2(M))$ by the C$^\ast$-algebra $\cK_M$ which is much larger than the ideal of compact operators on $L^2(M)$. This larger quotient turns out to provide von Neumann algebraic descriptions of the Akemann--Ostrand property~\cite{OzawaNote}. We use this to distinguish $M_q(\RH)$ and $M_0(\RH)$.

\subsection*{Acknowledgements}

The author thanks Mateusz Wasilewski, Changying Ding and Cyril Houdayer for comments on an earlier draft of this paper.


\section{Preliminaries}

$\cB(X, Y)$ denotes the bounded operators between Banach spaces $X \rightarrow Y$. $\mathcal{K}(X, Y)$ denotes the compact operators, meaning that they map the unit ball to a relatively compact set. We set $\cB(X) := \cB(X, X)$ and $\cK(X) := \cK(X, X)$.


The algebraic tensor product (vector space tensor product) is denoted by $\otimes_{\alg}$ and $\minotimes$ is the minimal tensor product of C$^\ast$-algebras. $\otimes$ is used for tensor products of elements.

We refer to~\cite{Takesaki1} as a standard reference on von Neumann algebras. For a von Neumann algebra $M$ we denote by $(M, L^2(M), J, L^2(M)^+)$ its standard form. For $x \in M$ we write $x^{\op} := J x^\ast J$ which is the right multiplication with $x$ on the standard space. For a finite von Neumann algebra $M$ with trace $\tau$ we have $M \subseteq L^2(M)$ where $L^2(M)$ is the completion of $M$ with respect to the inner product $\langle x, y \rangle = \tau(y^\ast x)$. Therefore every $T \in \cB(L^2(M))$ determines a map $Q_0(T) \in \cB(M, L^2(M))$ given by $x \mapsto T(x)$. Set
\[
Q_1: \cB(L^2(M)) \rightarrow \cB(M, L^2(M)) \slash \cK(M, L^2(M)): T \mapsto Q_0(T) + \cK(M, L^2(M)).
\]
$Q_1$ is clearly continuous and we define the closed left-ideal $\cK_{M}^L = \ker(Q_1)$ and the hereditary C$^\ast$-subalgebra $\cK_M = (\cK_{M}^L)^\ast \cap \cK_{M}^L$ of $\cB(L^2(M))$ (see also~\cite{OzawaNote}). We let $\mathcal{M}(\cK_M) \subseteq \cB(L^2(M))$ be the multiplier algebra of $\cK_M$; indeed this multiplier algebra is faithfully represented on $L^2(M)$ by~\cite[Proposition~2.1]{Lance}. Then $\cK_M$ is an ideal in the C$^\ast$-algebra $\mathcal{M}(\cK_M)$. We have $M \subseteq \mathcal{M}(\cK_M)$ and $M^\op \subseteq \mathcal{M}(\cK_M)$.

\subsection{A von Neumann version of the Akemann--Ostrand property}

\begin{defi}
Let $M$ be a finite von Neumann algebra. We say that $M$ has W$^\ast$AO if the map
\begin{equation}\label{Eqn=WAO}
\theta: M \otimes_{\alg} M^{\op} \rightarrow \mathcal{M}(\cK_M) \slash \cK_M: a \otimes b^{\op} \mapsto a b^{\op} + \cK_M.
\end{equation}
is continuous with respect to the minimal tensor norm and thus extends to a $\ast$-homomorphism $M \minotimes M^{\op} \rightarrow \mathcal{M}(\cK_M) \slash \cK_M$.
\end{defi}

We recall the following from~\cite[Section~4]{OzawaNote}. Let $\Gamma$ be a discrete group and let $\cL(\Gamma)$ and $\cR(\Gamma)$ be the left and right group von Neumann algebra respectively acting on $\ell^2(\Gamma)$. In this case $L^2(\cL(\Gamma)) \simeq \ell^2(\Gamma)$ as bimodules with the natural left and right actions of $\cL(\Gamma)$ and $\cR(\Gamma)$ on $\ell^2(\Gamma)$. We have $J \delta_s = \delta_{s^{-1}}$ which extends to an antilinear isometry on $\ell^2(\Gamma)$. Then $\cR(\Gamma) = J \cL(\Gamma) J$.

Assume $\Gamma$ is icc so that $\cL(\Gamma)$ and $\cR(\Gamma)$ are factors, i.e. $\cL(\Gamma) \cap \cR(\Gamma) = \mathbb{C} 1$. The map
\[
\pi: C^\ast(\cL(\Gamma), \cR(\Gamma)) \rightarrow \cB(\ell^2(\Gamma) \otimes \ell^2(\Gamma)): a b^{\op} \mapsto a \otimes b^{\op}, \qquad a \in \cL(\Gamma), b^{\op} \in \cR(\Gamma).
\]
is a well-defined $\ast$-homomorphism by Takesaki's theorem on minimality of the spatial tensor product. In~\cite[Section~4, Theorem]{OzawaNote} Ozawa showed the following theorem.

\begin{theo}\label{Thm=Ozawa}
Let $\Gamma$ be an exact icc group such that the $\ast$-homomorphism
\[
C_r^\ast(\Gamma) \otimes_{\alg} C_r^\ast(\Gamma)^{\op} \rightarrow \cB(\ell^2(\Gamma)) \slash \cK(\ell^2(\Gamma)): a \otimes b^{\op} \mapsto a b^{\op} + \cK(\ell^2(\Gamma)),
\]
is continuous with respect to $\minotimes$. Then $\cL(\Gamma)$ has W$^\ast$AO.
\end{theo}

\begin{proof}
By~\cite[Section~4, Theorem]{OzawaNote} we have
\[
\ker(\pi) = \mathcal{K}_{\cL(\Gamma)} \cap C^\ast(\cL(\Gamma), \cR(\Gamma)).
\]
Therefore,
\[
\cL(\Gamma) \minotimes \cR(\Gamma) \rightarrow\!\!\!\!\!^{\simeq \pi^{-1}} C^\ast(\cL(\Gamma), \cR(\Gamma)) \slash (\mathcal{K}_{\cL(\Gamma)} \cap C^\ast(\cL(\Gamma), \cR(\Gamma))),
\]
which concludes the theorem.
\end{proof}

\begin{rema}
It follows that if $\Gamma$ is an icc group that is bi-exact (or said to be in class $\mathcal{S}$, see~\cite[Section~15]{BrownOzawa}) then $\cL(\Gamma)$ has W$^\ast$AO.
\end{rema}

\subsection{\texorpdfstring{$q$}{q}-Gaussians}

Let $-1 < q < 1$ and let $\RH$ be a real Hilbert space with complexification $H := \RH \oplus i \RH$. Set the symmetrization operator $P_q^k$ on $H^{\otimes k}$,
\begin{equation}\label{Eqn=Pq}
P_q^k(\xi_1 \otimes \ldots \otimes \xi_n) = \sum_{\sigma \in S_k} q^{i(\sigma)} \xi_{\sigma(1)} \otimes \ldots \otimes \xi_{\sigma(n)},
\end{equation}
where $S_k$ is the symmetric group of permutations of $k$ elements and $i(\sigma) := \# \{ (a,b) \mid a < b,\linebreak \sigma(b) < \sigma(a) \}$ the number of inversions. The operator $P_q^k$ is positive and invertible~\cite{BS}. Define a new inner product on $H^{\otimes k}$ by
\[
\langle \xi, \eta \rangle_q := \langle P_q^k \xi, \eta \rangle,
\]
and call the new Hilbert space $H_q^{\otimes k}$. Set the Hilbert space direct sum $F_q(H) := \mathbb{C} \Omega \oplus (\bigoplus_{k=1}^\infty H_q^{\otimes k})$ where $\Omega$ is a unit vector called the vacuum vector. For $\xi \in H$ let
\[
l_q(\xi) (\eta_1 \otimes \ldots \otimes \eta_k) := \xi \otimes \eta_1 \otimes \ldots \otimes \eta_k, \qquad l_q(\xi) \Omega = \xi,
\]
and then $l_q^\ast(\xi) = l_q(\xi)^\ast$. These `creation' and `annihilation' operators are bounded and extend to $F_q(H)$. We define a von Neumann algebra by the double commutant
\[
M_q(\RH) := \{ l_q(\xi) + l_q^\ast(\xi) \mid \xi \in \RH \}''.
\]
Then $\tau_\Omega(x) := \langle x \Omega, \Omega \rangle$ is a faithful tracial state on $M_q(\RH)$ which is moreover normal. Now $F_q(H)$ is the standard form Hilbert space of $M_q(\RH)$ and $J x \Omega = x^\ast \Omega$. For vectors $\xi_1, \ldots, \xi_k \in H$ there exists a unique operator $W_q(\xi_1 \otimes \ldots \otimes \xi_k) \in M_q(\RH)$ such that
\[
W_q(\xi_1 \otimes \ldots \otimes \xi_k) \Omega = \xi_1 \otimes \ldots \otimes \xi_k.
\]
These operators are called Wick operators. It follows that $W_q(\xi)^\op \Omega = \xi$.

\begin{rema}\label{Remark=WAOZero}
Let $\mathbb{F}_\infty$ be the free group with countably infinitely many generators. $\mathbb{F}_\infty$ is icc and exact~\cite{BrownOzawa} and hence Theorem~\ref{Thm=Ozawa} applies. We conclude that $\cL(\mathbb{F}_\infty)$ has W$^\ast$AO. We have that $\cL(F_\infty) \simeq \Gamma_0(\RH)$ with $\RH$ a separable infinite dimensional real Hilbert space (see~\cite[Theorem~2.6.2]{DykVoic}) and so $\Gamma_0(\RH)$ has the W$^\ast$AO.
\end{rema}


\section{Non-isomorphism of \texorpdfstring{$q$}{q}-Gaussian von Neumann algebras}\label{Sect=qGaussian}

The following theorem provides a necessary condition for W$^\ast$AO.

\begin{theo}\label{Thm=AntiAO}
Let $M$ be a finite von Neumann algebra with finite normal faithful tracial state $\tau$. Suppose there exists a unital von Neumann subalgebra $B \subseteq M$ and infinitely many subspaces $M_i \subseteq M, i \in \mathbb{N}$ that are left and right $B$-invariant and mutually $\tau$-orthogonal in the sense that $\tau(y^\ast x) = 0$ for $x \in M_i, y \in M_j, i\not = j$. Suppose moreover that there exists $\delta > 0$ and finitely many operators $b_j, c_j \in B$, with $ \sum_j b_j \otimes c_j^{\op}$ non-zero, such that for every $i \in \mathbb{N}$ we have
\begin{equation}\label{Eqn=NonContraction}
\left\Vert Q_0\left(\sum_{j} b_j c_j^{op}\right) \right\Vert_{\cB(M_i, L^2(M_i))} \geq (1 + \delta) \left\Vert \sum_j b_j \otimes c_j^{\op} \right\Vert_{ B \minotimes B^{\op} }.
\end{equation}
Then $M$ does not have W$^\ast$AO.
\end{theo}

\begin{proof}
Let $X$ be the set of finite rank operators $x \in \cB(L^2(M))$ such that there exists $I_x \subseteq I$ finite with $\ker(x)^\perp \subseteq \oplus_{i \in I_x} L^2(M_i)$. Take $x \in X$ and choose $k \in I \backslash I_x$. Then,
\[
\begin{split}
\left\Vert Q_0\left(\sum_{j} b_j c_j^{op} + x\right) \right\Vert_{\cB(M, L_2(M)) }
&\geq \left\Vert Q_0\left(\sum_{j} b_j c_j^{op} + x\right) \right\Vert_{\cB(M_k, L^2(M_k)) } \\
&= \left\Vert Q_0\left(\sum_{j} b_j c_j^{op}\right) \right\Vert_{\cB(M_k, L^2(M_k)) } \\
&\geq (1 + \delta) \left\Vert \sum_j b_j \otimes c_j^{\op} \right\Vert_{B \minotimes B^{\op}}.
\end{split}
\]
The operators in $X$ are norm dense in $\cK(L^2(M))$ and by~\cite[Section~2, Proposition]{OzawaNote} we have that $Q_0(\cK(L^2(M)))$ is dense in $Q_0(\cK^L_M)$ in the norm of $\cB(M, L^2(M))$. As $Q_0$ is contractive $Q_0(X)$ is dense in $Q_0(\cK^L_M)$. It therefore follows that for any $x \in \cK_M^L$ we have
\[
\left\Vert Q_0\left(\sum_{j} b_j c_j^{op} + x\right) \right\Vert_{\cB(M, L_2(M)) } \geq (1 + \delta) \left\Vert \sum_j b_j \otimes c_j^{\op} \right\Vert_{B \minotimes B^{\op}}.
\]
Since $Q_0$ is contractive for every $x \in \cK_M^L$ we have,
\[
\left\Vert \sum_{j} b_j c_j^{op} + x \right\Vert_{\cB(L^2(M)) } \geq (1 + \delta) \left\Vert \sum_j b_j \otimes c_j^{\op} \right\Vert_{B \minotimes B^{\op}}.
\]
Hence, certainly for the Banach space quotient norm we have
\[
\left\Vert \sum_{j} b_j c_j^{op} + \cK_M \right\Vert_{\cB(L^2(M)) \slash \cK_M } \geq (1 + \delta) \left\Vert \sum_j b_j \otimes c_j^{\op} \right\Vert_{B \minotimes B^{\op}}.
\]
As the left hand side norm is the norm of the C$^\ast$-quotient $\mathcal{M}(\cK_M) \slash \cK_M$ this concludes the proof (see~\cite[Proposition~2.1]{Lance} as in the preliminaries).
\end{proof}

The proof of the following theorem essentially repeats its C$^\ast$-algebraic counterpart from~\cite[Theorem~3.3]{BCKW}.

\begin{theo}\label{Thm=AntiAOGaussian}
Assume $\dim(\RH) = \infty$ and $-1 < q < 1, q \not = 0$. Then the von Neumann algebra $M_q(\RH)$ does not have W$^\ast$AO.
\end{theo}

\begin{proof}
Let $d \geq 2$ be such that $q^2 d > 1$. Let
\[
M := M_q(\mathbb{R}^d \oplus \RH), \qquad B := M_q(\mathbb{R}^d \oplus 0).
\]
Let $\{ f_i \}_i$ be an infinite set of orthogonal vectors in $0 \oplus \RH$ such that $\Vert W_q(f_i) \Vert = 1$. Let $M_{q,i} := B W_q(f_i) B$ which is a $B$-$B$ invariant subset of $M$. Then $M_{q,i}$ and $M_{q,j}$ are $\tau_\Omega$-orthogonal if $i \not = j$. For $k \in \mathbb{N}$ let
\[
\cB(k) = \{ W_q(\xi) \mid \xi \in (\mathbb{R}^d \oplus 0)^{\otimes k} \}.
\]
It is proved in~\cite[Equation (3.2)]{BCKW} that for $b,c \in \cB(k)$ we have
\[
\langle b W_q(f_i) c \Omega, f_i \rangle_q = \langle b c^{\op} f_i, f_i \rangle_q = q^k \langle b c^{\op} \Omega, \Omega \rangle_q.
\]
Then for finitely many $b_j, c_j \in \cB(k)$ we have
\begin{multline}\label{Eqn=LowerEstimate}
\left\Vert Q_0\left(\sum_j b_j c_j^{\op}\right) \right\Vert_{\cB(M_{q,i}, L^2(M_{q,i}))}
\geq \left\Vert \sum_j b_j W_q(f_i) c_j \right\Vert_{ L^2(M_{q,i})}\\
\geq \left\vert \left\langle \sum_j b_j W_q(f_i) c_j \Omega, f_i \right\rangle_{\!q} \right\vert
= \left\vert \sum_j q^k \left\langle b_j \Omega c_j, \Omega \right\rangle_{\!q} \right\vert.
\end{multline}

Now let $\{ e_1, \ldots, e_d \} $ be an orthonormal basis of $\mathbb{R}^d \oplus 0$ and for $j = (j_1, \ldots, j_k) \in \{ 1,\ldots, d \}^k$ let $e_j = e_{j_1} \otimes \ldots \otimes e_{j_k}$. Let $J_k$ be the set of all such multi-indices of length $k$. So $\# J_k = d^ k$. Set $\xi_j = (P^k_q)^{- \frac{1}{2} } e_j$ so that $\langle \xi_j, \xi_j \rangle_q = \langle P_q^k \xi_j, \xi_j \rangle = 1$.

Now~\eqref{Eqn=LowerEstimate} yields that for all $k \geq 1$ and all $i$,
\[
\begin{split}
\left\Vert Q_0\left(\sum_{j \in J_k} W_q(\xi_j)^\ast W_q(\xi_j)^{\op}\right) \right\Vert_{\cB(M_{q,i}, L^2(M_{q,i}))}
&\geq \sum_{j \in J_k} q^k \langle W_q(\xi_j)^\ast \Omega W_q(\xi_j), \Omega \rangle_q\\
&= \sum_{j \in J_k} q^k \langle \Omega W_q(\xi_j), W_q(\xi_j) \Omega \rangle_q\\
&= \sum_{j \in J_k} q^k \langle \xi_j, \xi_j \rangle_q = q^k d^{k}.
\end{split}
\]
From~\cite[Proof of Theorem~2]{Nou} (or see~\cite[Proof of Theorem~3.3]{BCKW}) we find,
\[
\left\Vert \sum_{j \in J_k} W_q(\xi_j)^\ast \otimes W_q(\xi_j)^{\op} \right\Vert_{B \minotimes B^{\op}} \leq \left(\prod_{i=1}^\infty (1 - q^i)^{-1}\right)^3 (k+1)^2 d^{k/2}.
\]
Therefore, as $q^2 d > 1$ there exists $\delta > 0$ such that for $k$ large enough we have for every $i$,
\[
\left\Vert Q_0\left(\sum_{j \in J_k} W_q(\xi_j)^\ast W_q(\xi_j)^{\op}\right) \right\Vert_{\cB(M_{q,i}, L^2(M_{q,i})) } \geq (1+ \delta) \left\Vert \sum_{j \in J_k} W_q(\xi_j)^\ast \otimes W_q(\xi_j)^{\op} \right\Vert_{B \minotimes B^{\op}}.
\]
Hence the assumptions of Theorem~\ref{Thm=AntiAO} are witnessed which shows that W$^\ast$AO does not hold.
\end{proof}

\begin{coro}
Let $\RH$ be an infinite dimensional real separable Hilbert space. The von Neumann algebras $\Gamma_0(\RH)$ and $\Gamma_q(\RH)$ with $-1 < q < 1, q \not = 0$ are non-isomorphic.
\end{coro}

\begin{proof}
This is a consequence of Theorem~\ref{Thm=AntiAOGaussian} and Remark~\ref{Remark=WAOZero} as W$^\ast$AO is preserved under isomorphism.
\end{proof}

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