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

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 $-1<q<1,q\ne 0$ 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].

Proceedings of the AMS, to appear