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تسجيل مستخدم جديدProperty (T), vanishing cohomology, and strong $1$-boundedness
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The notion of strong 1-boundedness for finite von Neumann algebras was introduced by Jung. 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. We also give a new proof of a result of Shlyakhtenko which states that if $G$ is a sofic, finitely presented group with vanishing first $ell^2$-Betti number, then $L(G)$ is strongly 1-bounded. 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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