Compact actions and uniqueness of the group measure space decomposition of II${}_1$ factors

Abstract

We prove that any II${}_1$ factor $L^{\infty }(X)⋊\Gamma$ arising from a compact, free, ergodic, probability measure preserving action of a countable group $\Gamma$ with positive first ${\ell }^2$-Betti number, has a unique group measure space Cartan subalgebra, up to unitary conjugacy.