Topological dynamical systems associated to II 1-factors

Abstract

If N ⊂ R ω is a separable II 1-factor, the space H om ( N , R ω ) of unitary equivalence classes of unital ⁎-homomorphisms N → R ω is shown to have a surprisingly rich structure. If N is not hyperfinite, H om ( N , R ω ) 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 H om ( N , R ω ) 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 H om ( N , R ω ) , 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.