Normalizers inside Amalgamated Free Product von Neumann Algebras

Abstract

Recently, Adrian Ioana proved that all crossed products $L^{\infty }(X)⋊({\Gamma }_1\ast {\Gamma }_2)$ by free ergodic probability measure preserving actions of a nontrivial free product group ${\Gamma }_1\ast {\Gamma }_2$ have a unique Cartan subalgebra up to unitary conjugacy. Ioana deduced this result from a more general dichotomy theorem on the normalizer ${\mathcal{N}}_M(A{)}^{\prime \prime }$ of an amenable subalgebra $A$ of an amalgamated free product von Neumann algebra $M=M_1{\ast }_B{M}_2$. We improve this dichotomy theorem by removing the spectral gap assumptions and obtain in particular a simpler proof for the uniqueness of the Cartan subalgebra in $L^{\infty }(X)⋊({\Gamma }_1\ast {\Gamma }_2)$.