# 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

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