Compact actions and uniqueness of the group measure space decomposition of II1 factors

We prove that any II1 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 (2) 1 (Γ ) > 0, has a unique group measure space Cartan subalgebra, up to unitary conjugacy. © 2012 Elsevier Inc. All rights reserved.