On the isomorphism class of $q$-Gaussian W$^\ast$-algebras for infinite variables

TL;DR

This paper proves that $q$-Gaussian von Neumann algebras for infinite-dimensional spaces are not isomorphic to the free case when $q eq 0$, extending previous C*-algebra results to the von Neumann algebra setting.

Contribution

It establishes the non-isomorphism of $q$-Gaussian von Neumann algebras with the free case for infinite variables, using Ozawa's ideas.

Findings

$q$-Gaussian von Neumann algebras are not isomorphic to the free case for $q eq 0$

The non-isomorphism extends from C*-algebras to von Neumann algebras

The result applies to infinite-dimensional real Hilbert spaces

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 [BCKW22]. Using ideas of Ozawa we show that this non-isomorphism result also holds on the level of von Neumann algebras.