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

TL;DR

This paper investigates the isomorphism classes of $q$-Gaussian C$^ ext{*}$-algebras for infinite-dimensional real Hilbert spaces, showing that for non-zero $q$, these algebras are not isomorphic to the $q=0$ case, revealing structural differences.

Contribution

It proves that for infinite-dimensional spaces and non-zero $q$, the $q$-Gaussian C$^ ext{*}$-algebras are not isomorphic to the $q=0$ case, answering open questions in the field.

Findings

$M_q(H_{ ext{R}})$ lacks the Akemann-Ostrand property for $q eq 0$

$A_q(H_{ ext{R}})$ is not isomorphic to $A_0(H_{ ext{R}})$ for infinite dimensions and $q eq 0$

Provides a partial answer to open questions in the literature

Abstract

For a real Hilbert space $H_{\mathbb{R}}$ and $-1 < q < 1$ Bozejko and Speicher introduced the C$^\ast$-algebra $A_q(H_{\mathbb{R}})$ and von Neumann algebra $M_q(H_{\mathbb{R}})$ of $q$-Gaussian variables. We prove that if $\dim(H_{\mathbb{R}}) = \infty$ and $-1 < q < 1, q \not = 0$ then $M_q(H_{\mathbb{R}})$ does not have the Akemann-Ostrand property with respect to $A_q(H_{\mathbb{R}})$. It follows that $A_q(H_{\mathbb{R}})$ is not isomorphic to $A_0(H_{\mathbb{R}})$. This gives an answer to the C$^\ast$-algebraic part of Question 1.1 and Question 1.2 in [NeZe18].