# Journal of Operator Theory

Volume 81, Issue 1, Winter 2019  pp. 107-132.

A note on relative amenability of finite von Neumann algebras

Authors:  Xiaoyan Zhou (1), Junsheng Fang (2)
Author institution: (1) School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China
(2) School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

Summary:  Let $M$ be a finite von Neumann algebra (respectively, a type II$_{1}$ factor) and let $N\subset M$ be a II$_{1}$ factor (respectively, $N\subset M$ have an atomic part). We prove that if the inclusion $N\subset M$ is amenable, then 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.

DOI: http://dx.doi.org/10.7900/jot.2017dec06.2200
Keywords:  II$_1$ factors, finite von Neumann algebras, relative amenability, trace preserving normal unital completely positive maps, Haagerup property, weak Haagerup property, weak exactness