Uniqueness of the group measure space decomposition for Popa's $\Cal H\Cal T$ factors

Abstract

We prove that every group measure space II${}_1$ factor $L^{\infty }(X)⋊\Gamma$ coming from a free ergodic rigid (in the sense of [Po01]) probability measure preserving action of a group $\Gamma$ with positive first ${\ell }^2$--Betti number, has a unique group measure space Cartan subalgebra, up to unitary conjugacy. We deduce that many $\Cal H\Cal T$ factors, including the II${}_1$ factors associated with the actions $\Gamma \curvearrowright {\mathbb{T}}^2$ and $\Gamma \curvearrowright$ SL${}_2(\mathbb{R})$/SL${}_2(\mathbb{Z})$, where $\Gamma$ is a non--amenable subgroup of SL${}_2(\mathbb{Z})$, have a unique group measure space Cartan subalgebra.

improved exposition

References in corpus (2)

• Some unique group-measure space decomposition results
• Examples of group actions which are virtually W*-superrigid