Amenable absorption in amalgamated free product von Neumann algebras

TL;DR

This paper studies how amenable subalgebras behave within amalgamated free product von Neumann algebras, showing that large intersections with one component imply containment, using non-normal conditional expectations.

Contribution

It establishes a new amenable absorption property in amalgamated free product von Neumann algebras without relying on Popa's asymptotic orthogonality.

Findings

Amenable subalgebras with large intersection are contained in the component algebra.

The proof introduces techniques based on non-normal conditional expectations.

Results do not depend on Popa's asymptotic orthogonality property.

Abstract

We investigate the position of amenable subalgebras in arbitrary amalgamated free product von Neumann algebras $M = M_1 \ast_B M_2$. Our main result states that under natural analytic assumptions, any amenable subalgebra of $M$ that has a large intersection with $M_1$ is actually contained in $M_1$. The proof does not rely on Popa's asymptotic orthogonality property but on the study of non normal conditional expectations.