A note on relative amenable of finite von Neumann algebras

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.