On the isomorphism class of 𝑞-Gaussian C*-algebras for infinite variables

Borst, Matthijs; Caspers, Martijn; Klisse, Mario; Wasilewski, Mateusz

doi:10.1090/proc/16165

On the isomorphism class of $q$-Gaussian C$^\ast$-algebras for infinite variables
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by Matthijs Borst, Martijn Caspers, Mario Klisse and Mateusz Wasilewski;

Proc. Amer. Math. Soc. 151 (2023), 737-744

DOI: https://doi.org/10.1090/proc/16165

Published electronically: November 22, 2022

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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 raised by Nelson and Zeng [Int. Math. Res. Not. IMRN 17 (2018), pp. 5486–5535].

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