# A note on relative amenable of finite von Neumann algebras

## TL;DR

This paper investigates the properties of inclusions of finite von Neumann algebras, showing that amenability of the inclusion implies an approximate factorization of the identity map, and establishes permanence properties like weak Haagerup and weak exactness.

## Contribution

It generalizes Haagerup's result by proving approximate factorizations for amenable inclusions and demonstrates their permanence properties.

## Key findings

- Amenable inclusion implies approximate factorization of the identity map.
- Establishes permanence of weak Haagerup property.
- Proves weak exactness for amenable inclusions.

## Abstract

Let $M$ be a finite von Neumann algebra (resp. a type II$_{1}$ factor) and let $N\subset M$ be a II$_{1}$ factor (resp. $N\subset M$ have an atomic part). We prove that the inclusion $N\subset M$ is amenable implies the identity map on $M$ has an approximate factorization through $M_m(\mathbb{C})\otimes N $ via trace preserving normal unital completely positive maps, which is a generalization of a result of Haagerup. We also prove two permanence properties for amenable inclusions. One is weak Haagerup property, the other is weak exactness.