# Any generating set of an arbitrary property T von Neumann algebra has free <br>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, δ0(X), is less than or equal to 1. It follows that when N embeds into the ultraproduct of the hyperfinite II1 factor, then δ0(X) = 1 and otherwise, δ0(X) = −∞. This generalizes the earlier results of Voiculescu, and Ge, Shen pertaining to SLn(ℤ) as well as the results of Connes, Shlyakhtenko pertaining to group generators of arbitrary property T algebras.