A note on characterizations of relative amenability on finite von Neumann algebras

TL;DR

This paper introduces new characterizations of relative amenability in finite von Neumann algebras and demonstrates a stable property of such inclusions, linking local and global amenability.

Contribution

It provides two novel characterizations of relative amenability and establishes a stability result for amenable inclusions under certain conditions.

Findings

New characterizations of relative amenability

Stable property of amenable inclusions

Equivalence of local and global amenability under assumptions

Abstract

In this paper, we give another two characterizations of relative amenability on finite von Neumann algebras, one of which can be thought of as an analogue of injective operator systems. As an application, we prove a stable property of relative amenable inclusions. We prove that under certain assumptions, the inclusion $N=\int_{X} \bigoplus N_{p} d \mu\subset M=\int_{X} \bigoplus M_{p} d \mu$ is amenable if and only if $N_p\subset M_p$ is amenable almost everywhere.