Unique Cartan decomposition for II_1 factors arising from arbitrary actions of free groups

Details Unique Cartan decomposition for II_1 factors arising from arbitrary actions of free groups Unique Cartan decomposition for II_1 factors arising from arbitrary actions of free groups

2011-11-29

arxiv.org/abs/1111.6951v2

Mathematics

Operator Algebras

v2: improvement of a number of results; comments on the proofs added

Scientific paper

We prove that for any free ergodic probability measure preserving action \F_n \actson (X,\mu) of a free group on n generators \F_n, 2 \leq n \leq \infty, the associated group measure space II_1 factor $L^\infty(X) \rtimes \F_n$ has L^\infty(X) as its unique Cartan subalgebra, up to unitary conjugacy. We deduce that group measure space II_1 factors arising from actions of free groups with different number of generators are never isomorphic. We actually prove unique Cartan decomposition results for II_1 factors arising from arbitrary actions of a rather large family of groups, including all free products of amenable groups and their direct products.

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