Topological dynamical systems associated to II1-factors

If N⊂Rω is a separable II1-factor, the space Hom(N,Rω) of unitary equivalence classes of unital ⁎-homomorphisms N→Rω is shown to have a surprisingly rich structure. If N is not hyperfinite, Hom(N,Rω) is an infinite-dimensional, complete, metrizable topological space with convex-like structure, and the outer automorphism group Out(N) acts on it by “affine” homeomorphisms. (If N≅R, then Hom(N,Rω) is just a point.) Property (T) is reflected in the extreme points – theyʼre discrete in this case. For certain free products N=Σ⁎R, every countable group acts nontrivially on Hom(N,Rω), 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.