A note on relative amenability of finite von Neumann algebras

Abstract

Let M be a finite von Neumann algebra (respectively, a type II1 factor) and let N⊂M be a II1 factor (respectively, N⊂M have an atomic part). We prove that if the inclusion N⊂M is amenable, then implies the identity map on M has an approximate factorization through Mm(C)⊗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.