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 is a diffuse, property T von Neumann algebra and is an arbitrary finite generating set of selfadjoint elements for . By using rigidity/deformation arguments applied to representations of in ultraproducts of full matrix algebras, we deduce that the microstate spaces of are asymptotically discrete up to unitary conjugacy. We use this description to show that the free entropy dimension of , , is less than or equal to . It follows that when embeds into the ultraproduct of the hyperfinite factor, then and otherwise, . This generalizes the earlier results of Voiculescu, and Ge, Shen pertaining to 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