All generating sets of all property T von Neumann algebras have free entropy dimension $\le 1$

Abstract

Suppose $N$ is a diffuse, property T von Neumann algebra and X is an arbitrary finite generating set of selfadjoint elements for N. By using rigidity/deformation arguments applied to representations of N in full matrix algebras, we deduce that the microstate spaces of X are asymptotically discrete up to unitary conjugacy. We use this description to show that the free entropy dimension of X, ${\delta }_0(X)$, is less than or equal to 1. It follows that when N embeds into the ultraproduct of the hyperfinite ${\mathrm{II}}_1$-factor, then ${\delta }_0(X)=1$ and otherwise, $δ_0(X)=-\infinity$. This generalizes the earlier results of Voiculescu, and Ge, Shen pertaining to $S{L}_n(\mathbb{Z})$ as well as the results of Connes, Shlyakhtenko pertaining to group generators of arbitrary property T algebras.

6 pages

References in corpus (1)

• Free probability, free entropy and applications to von Neumann algebras

Cited by in corpus (2)

• Topological Free Entropy Dimension in Unital C^* algebras (II) : Orthogonal Sum of Unital C^*-algebras
• The free entropy dimension of some $L^{\infty }[0,1]$-circular operators