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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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

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].
References
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Bibliographic Information
  • Matthijs Borst
  • Affiliation: TU Delft, EWI/DIAM, P.O. Box 5031, 2600 Georgia Delft, The Netherlands
  • MR Author ID: 1440742
  • ORCID: 0000-0003-4550-5099
  • Email: m.j.borst@tudelft.nl
  • Martijn Caspers
  • Affiliation: TU Delft, EWI/DIAM, P.O. Box 5031, 2600 Georgia Delft, The Netherlands
  • MR Author ID: 880229
  • Email: m.p.t.caspers@tudelft.nl
  • Mario Klisse
  • Affiliation: TU Delft, EWI/DIAM, P.O. Box 5031, 2600 Georgia Delft, The Netherlands
  • MR Author ID: 1402817
  • ORCID: 0000-0003-0246-619X
  • Email: m.klisse@tudelft.nl
  • Mateusz Wasilewski
  • Affiliation: Mateusz Wasilewski, IMPAN, Śniadeckich 8, 00-656 Warsaw, Poland
  • MR Author ID: 1146862
  • ORCID: 0000-0002-3952-777X
  • Email: mwasilewski@impan.pl
  • Received by editor(s): February 28, 2022
  • Received by editor(s) in revised form: May 13, 2022
  • Published electronically: November 22, 2022
  • Additional Notes: The second author was supported by the NWO Vidi grant VI.Vidi.192.018 ‘Non-commutative harmonic analysis and rigidity of operator algebras’.
    The third author was supported by the NWO project 613.009.125 ‘The structure of Hecke-von Neumann algebras’.
    The fourth author was supported by the European Research Council Starting Grant 677120 INDEX
  • Communicated by: Adrian Ioana
  • © Copyright 2022 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 151 (2023), 737-744
  • MSC (2020): Primary 46L35, 46L06
  • DOI: https://doi.org/10.1090/proc/16165
  • MathSciNet review: 4520022