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PROPERTY (T) AND STRONG 1-BOUNDEDNESS FOR VON NEUMANN ALGEBRAS

Part of: Selfadjoint operator algebras Locally compact groups and their algebras Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  01 September 2025

Ben Hayes Open the ORCID record for Ben Hayes [Opens in a new window] ,
David Jekel Open the ORCID record for David Jekel [Opens in a new window]  and
Srivatsav Kunnawalkam Elayavalli Open the ORCID record for Srivatsav Kunnawalkam Elayavalli [Opens in a new window]
Ben Hayes
Affiliation:
Department of Mathematics, University of Virginia, 141 Cabell Drive, Kerchof Hall, P.O. Box 400137 Charlottesville, VA 22904 (brh5c@virginia.edu)
David Jekel
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 København Ø, Denmark (daj@math.ku.dk)
Srivatsav Kunnawalkam Elayavalli*
Affiliation:
Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Station B 407807, Nashville, TN 37240
*
srivatsav.kunnawalkam.elayavalli@vanderbilt.edu
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Abstract

The notion of strong 1-boundedness for finite von Neumann algebras was introduced in [Jun07b]. This framework provided a free probabilistic approach to study rigidity properties and classification of finite von Neumann algebras. In this paper, we prove that tracial von Neumann algebras with a finite Kazhdan set are strongly 1-bounded. This includes all property (T) von Neumann algebras with finite-dimensional center and group von Neumann algebras of property (T) groups. This result generalizes all the previous results in this direction due to Voiculescu, Ge, Ge-Shen, Connes-Shlyakhtenko, Jung-Shlyakhtenko, Jung and Shlyakhtenko. Our proofs are based on analysis of covering estimates of microstate spaces using an iteration technique in the spirit of Jung.

Keywords

Property Tstrongly 1-boundedfree entropy dimension

MSC classification

Primary: 46L10: General theory of von Neumann algebras 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations 46B06: Asymptotic theory of Banach spaces

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , First View , pp. 1 - 34
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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Footnotes

B. Hayes gratefully acknowledges support from the NSF grant DMS-2000105. D. Jekel was supported by NSF grant DMS-2002826

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