Journal of Operator Theory

Volume 85, Issue 1, Winter 2021 pp. 217-228.

Von Neumann algebras of sofic groups with $\beta_{1}^{(2)}=0$ are strongly $1$ -bounded

Authors: Dimitri Shlyakhtenko
Author institution: Department of Mathematics, UCLA, Los Angeles, CA 90095, U.S.A.

Summary: We show that if $\Gamma$ is a finitely generated finitely presented sofic group with zero first $L^{2}$ -Betti number, then the von Neumann algebra $L(\Gamma)$ is strongly $1$ -bounded in the sense of Jung. In particular, $L(\Gamma)\not\cong L(\Lambda)$ if $\Lambda$ is any group with free entropy dimension $>1$ , for example a free group. The key technical result is a short proof of an estimate of Jung

DOI: http://dx.doi.org/10.7900/jot.2019oct21.2270
Keywords: free probability, free entropy, $L^2$ -Betti numbers

Contents Full-Text PDF