Compact actions and uniqueness of the group measure space decomposition of II ~1 factors

摘要

We prove that any II _1 factor L ~∞(X)?Γ arising from a compact, free, ergodic, probability measure preserving action of a countable group Γ with positive first ? ~2-Betti number, β _1 ~((2))(Γ)>0, has a unique group measure space Cartan subalgebra, up to unitary conjugacy.