Normalizers inside amalgamated free product von Neumann algebras

TL;DR

This paper improves a dichotomy theorem for amalgamated free product von Neumann algebras, removing spectral gap assumptions, and simplifies the proof of the uniqueness of Cartan subalgebras in crossed products by free product groups.

Contribution

It removes spectral gap assumptions from the dichotomy theorem, leading to a simpler proof of Cartan subalgebra uniqueness in free product group actions.

Findings

Enhanced dichotomy theorem without spectral gap assumptions

Simplified proof for Cartan subalgebra uniqueness

Broader applicability to free product group actions

Abstract

Recently, Adrian Ioana proved that all crossed products by free ergodic probability measure preserving actions of a nontrivial free product group \Gamma_1 * \Gamma_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 crossed products by \Gamma_1 * \Gamma_2.