# Normalizers inside Amalgamated Free Product von Neumann Algebras

### Stefaan Vaes

Katholieke Universiteit Leuven, Belgium

## Abstract

Recently, Adrian Ioana proved that all crossed products $L_{∞}(X)⋊(Γ_{1}∗Γ_{2})$ by free ergodic probability measure preserving actions of a nontrivial free product group $Γ_{1}∗Γ_{2}$ have a unique Cartan subalgebra up to unitary conjugacy. Ioana deduced this result from a more general dichotomy theorem on the normalizer $N_{M}(A)_{′′}$ of an amenable subalgebra $A$ of an amalgamated free product von Neumann algebra $M=M_{1}∗_{B}M_{2}$. 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 $L_{∞}(X)⋊(Γ_{1}∗Γ_{2})$.

## Cite this article

Stefaan Vaes, Normalizers inside Amalgamated Free Product von Neumann Algebras. Publ. Res. Inst. Math. Sci. 50 (2014), no. 4, pp. 695–721

DOI 10.4171/PRIMS/147