Group measure space decomposition of II_1 factors and W*-superrigidity

Details Group measure space decomposition of II_1 factors and W*-superrigidity Group measure space decomposition of II_1 factors and W*-superrigidity

2009-06-15

arxiv.org/abs/0906.2765v2

Inventiones Mathematicae 182 (2010), 371-417

Mathematics

Operator Algebras

Final version. Some extra details have been added to improve the exposition

Scientific paper

We prove a "unique crossed product decomposition" result for group measure space II_1 factors arising from arbitrary free ergodic probability measure preserving (p.m.p.) actions of groups \Gamma in a fairly large family G, which contains all free products of a Kazhdan group and a non-trivial group, as well as certain amalgamated free products over an amenable subgroup. We deduce that if T_n denotes the group of upper triangular matrices in PSL(n,Z), then any free, mixing p.m.p. action of the amalgamated free product of PSL(n,Z) with itself over T_n, is W*-superrigid, i.e. any isomorphism between L^\infty(X) \rtimes \Gamma and an arbitrary group measure space factor L^\infty(Y) \rtimes \Lambda, comes from a conjugacy of the actions. We also prove that for many groups \Gamma in the family G, the Bernoulli actions of \Gamma are W*-superrigid.

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