# Any generating set of an arbitrary property T von Neumann algebra has free entropy dimension ≤ 1

### Kenley Jung

UCLA### Dimitri L. Shlyakhtenko

UCLA

## Abstract

Suppose that $N$ 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 ultraproducts of 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 $X$, $δ_{0}(X)$, is less than or equal to $1$. It follows that when $N$ embeds into the ultraproduct of the hyperfinite $II_{1}$ factor, then $δ_{0}(X)=1$ and otherwise, $δ_{0}(X)=−∞$. This generalizes the earlier results of Voiculescu, and Ge, Shen pertaining to $SL_{n}(Z)$ as well as the results of Connes, Shlyakhtenko pertaining to group generators of arbitrary property T algebras.

## Cite this article

Kenley Jung, Dimitri L. Shlyakhtenko, Any generating set of an arbitrary property T von Neumann algebra has free entropy dimension ≤ 1. J. Noncommut. Geom. 1 (2007), no. 2, pp. 271–279

DOI 10.4171/JNCG/7