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

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 $X$, ${\delta }_0(X)$, 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, ${\delta }_0(X)=-\infty$. This generalizes the earlier results of Voiculescu, and Ge, Shen pertaining to ${\mathrm{SL}}_n(\mathbb{Z})$ as well as the results of Connes, Shlyakhtenko pertaining to group generators of arbitrary property T algebras.