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

Abstract

For a real Hilbert space H R H_{\mathbb {R}} and − 1 > q > 1 -1 > q > 1 Bozejko and Speicher introduced the C ∗ ^\ast -algebra A q ( H R ) A_q(H_{\mathbb {R}}) and von Neumann algebra M q ( H R ) M_q(H_{\mathbb {R}}) of q q -Gaussian variables. We prove that if dim ⁡ ( H R ) = ∞ \dim (H_{\mathbb {R}}) = \infty and − 1 > q > 1 , q ≠ 0 -1 > q > 1, q \not = 0 then M q ( H R ) M_q(H_{\mathbb {R}}) does not have the Akemann-Ostrand property with respect to A q ( H R ) A_q(H_{\mathbb {R}}) . It follows that A q ( H R ) A_q(H_{\mathbb {R}}) is not isomorphic to A 0 ( H R ) 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].