Topological dynamical systems associated to II_1 factors

Abstract

If $N\subset \mathbb{R}$ is a separable II${}_1$-factor, the space $\Hom(N,\R)$ of unitary equivalence classes of unital *-homomorphisms $N\to \mathbb{R}$ is shown to have a surprisingly rich structure. If $N$ is not hyperfinite, $\Hom(N,\R)$ is an infinite-dimensional, complete, metrizeable topological space with convex-like structure, and the outer automorphism group $\Out(N)$ acts on it by "affine" homeomorphisms. (If $N\cong R$, then $\Hom(N,\R)$ is just a point.) Property (T) is reflected in the extreme points -- they're discrete in this case. For certain free products $N=\Sigma \ast R$, every countable group acts nontrivially on $\Hom(N, \R)$, and we show the extreme points are not discrete for these examples. Finally, we prove that the dynamical systems associated to free group factors are isomorphic.

30 pages, including an appendix written by Narutaka Ozawa, this version corrects a misattribution in the published paper (and I've added the proper reference to a paper of Popa); published in Adv. Math. (2011)