Topological dynamical systems associated to II1-factors

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If N⊂Rω is a separable II1-factor, the space H{double-struck}om(N,Rω) 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,Rω) 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,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.

Original languageEnglish (US)
Pages (from-to)1665-1699
Number of pages35
JournalAdvances in Mathematics
Issue number4
StatePublished - Jul 10 2011

All Science Journal Classification (ASJC) codes

  • General Mathematics


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