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On the isomorphism class of q-Gaussian W * -algebras for infinite variables
Martijn Caspers1
1 TU Delft, EWI/DIAM, P.O.Box 5031, 2600 GA Delft, The Netherlands
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1711-1716.

Let M q (H ℝ ) be the q-Gaussian von Neumann algebra associated with a separable infinite dimensional real Hilbert space H ℝ where -1<q<1. We show that M q (H ℝ )¬≃M 0 (H ℝ ) for -1<q≠0<1. The C * -algebraic counterpart of this result was obtained recently in [1]. Using ideas of Ozawa we show that this non-isomorphism result also holds on the level of von Neumann algebras.

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DOI : 10.5802/crmath.489
Classification : 46L35, 46L06
Mots clés : $q$-Gaussian von Neumann algebras, Akemann–Ostrand property
Martijn Caspers 1

1 TU Delft, EWI/DIAM, P.O.Box 5031, 2600 GA Delft, The Netherlands
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {On the isomorphism class of $q${-Gaussian} {W}$^\ast $-algebras for infinite variables},
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Martijn Caspers. On the isomorphism class of $q$-Gaussian W$^\ast $-algebras for infinite variables. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1711-1716. doi : 10.5802/crmath.489. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.489/

[1] Matthijs Borst; Martijn Caspers; Mario Klisse; Mateusz Wasilewski On the isomorphism class of q-Gaussian C * -algebras for infinite variables (2022) (to appear in Proc. Am. Math. Soc.) | arXiv

[2] Marek Bożejko; Burkhard Kümmerer; Roland Speicher q-Gaussian processes: non-commutative and classical aspects, Commun. Math. Phys., Volume 185 (1997) no. 1, pp. 129-154 | MR | Zbl

[3] Marek Bożejko; Roland Speicher An example of a generalized Brownian motion, Commun. Math. Phys., Volume 137 (1991) no. 3, pp. 519-531 | DOI | MR | Zbl

[4] Nathanial P. Brown; Narutaka Ozawa C * -algebras and finite-dimensional approximations, Graduate Studies in Mathematics, 88, American Mathematical Society, 2008, xvi+509 pages

[5] Yoann Dabrowski A free stochastic partial differential equation, Ann. Inst. Henri Poincaré, Volume 50 (2014) no. 4, pp. 1404-1455 | Numdam | MR | Zbl

[6] Changying Ding; Srivatsav Kunnawalkam Elayavalli; Jesse Peterson Properly Proximal von Neumann Algebras (2022) | arXiv

[7] Kenneth J. Dykema; Alexandru Nica; Dan V. Voiculescu Free Random Variables, CRM Monograph Series, 1, American Mathematical Society, 1992, v+70 pages

[8] Alice Guionnet; Dimitri Shlyakhtenko Free monotone transport, Invent. Math., Volume 197 (2014) no. 3, pp. 613-661 | DOI | MR

[9] E. Christopher Lance Hilbert C * -modules, A toolkit for operator algebraists, London Mathematical Society Lecture Note Series, 20, London Mathematical Society, 1995, x+130 pages | DOI

[10] Brent Nelson; Qiang Zeng Free monotone transport for infinite variables, Int. Math. Res. Not., Volume 2018 (2018) no. 17, pp. 5486-5535 | DOI | MR | Zbl

[11] Alexandre Nou Non injectivity of the q-deformed von Neumann algebra, Math. Ann., Volume 330 (2004) no. 1, pp. 17-38 | MR | Zbl

[12] Narutaka Ozawa A comment on free group factors, Noncommutative harmonic analysis with applications to probability. II (Banach Center Publications), Volume 89, Polish Academy of Sciences, 2010, pp. 241-245 | DOI | MR | Zbl

[13] Masamichi Takesaki Theory of operator algebras. I, Springer, 1979, vii+415 pages | DOI

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