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

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Isomorphism (crystallography) Combinatorics Hilbert space Mathematics Space (punctuation) Algebraic number Algebra over a field Discrete mathematics Physics Pure mathematics Crystallography Mathematical analysis Crystal structure Philosophy Linguistics Chemistry

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

YOU? · · 2022 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2202.13640 · OA: W4221157366

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].