PROPERTY (T) AND STRONG 1-BOUNDEDNESS FOR VON NEUMANN ALGEBRAS
Ben Hayes, David Jekel, Srivatsav Kunnawalkam Elayavalli
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Source: Crossref
Published: Sep 1, 2025
DOI: 10.1017/s1474748024000446
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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.
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