Topological dynamical systems associated to II_1 factors

If $N\subset \mathbb{R}$ is a separable II${}_1$-factor, the space $\text{Hom}(N,\mathbb{R})$ of unitary equivalence classes of unital *-homomorphisms $N\to \mathbb{R}$ is shown to have a surprisingly rich structure. If $N$ is not hyperfinite, $\text{Hom}(N,\mathbb{R})$ is an infinite-dimensional, complete, metrizeable topological space with convex-like structure, and the outer automorphism group $\text{Out}(N)$ acts on it by “affine” homeomorphisms. (If $N\cong R$, then $\text{Hom}(N,\mathbb{R})$ is just a point.) Property (T) is reflected in the extreme points – they’re discrete in this case. For certain free products $N=\mathrm{\Sigma }\ast R$, every countable group acts nontrivially on $\text{Hom}(N,\mathbb{R})$, 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.