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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