All generating sets of all property T von Neumann algebras have free
entropy dimension $\leq 1$
Authors:
Kenley Jung,
Dimitri Shlyakhtenko
Comments: 6 pages
Subj-class: Operator Algebras
MSC-class: 46L54; 52C17
Suppose $N$ 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 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, $\delta_0(X)=-\infinity$. This generalizes the
earlier results of Voiculescu, and Ge, Shen pertaining to $SL_n(\mathbb Z)$ as
well as the results of Connes, Shlyakhtenko pertaining to group generators of
arbitrary property T algebras.
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