Topological dynamical systems associated to II₁-factors

If N⊂R^ω is a separable II₁-factor, the space H{double-struck}om(N,R^ω) of unitary equivalence classes of unital *-homomorphisms N→R; is shown to have a surprisingly rich structure. If N is not hyperfinite, Hom(N,R) 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 NR, then Hom(N,R^ω) is just a point.) Property (T) is reflected in the extreme points - they're discrete in this case. For certain free products N=ς*R, every countable group acts nontrivially on H{double-struck}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.