TY - JOUR

T1 - Topological dynamical systems associated to II1-factors

AU - Brown, Nathanial P.

N1 - Funding Information:
1 Supported by NSF grants DMS-0554870 and DMS-0856197.

PY - 2011/7/10

Y1 - 2011/7/10

N2 - 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.

AB - 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.

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U2 - 10.1016/j.aim.2011.04.003

DO - 10.1016/j.aim.2011.04.003

M3 - Article

AN - SCOPUS:79955961450

VL - 227

SP - 1665

EP - 1699

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 4

ER -