Free Entropy Dimension and Property T

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) = −∞.