Journal of Operator Theory

Volume 65, Issue 2, Spring 2011 pp. 403-417.

Some remarks on Haagerup's approximation property

Authors: Jon P. Bannon (1) and Junsheng Fang (2)
Author institution: (1) Department of Mathematics, Siena College, Loudonville NY, 12065, U.S.A.
(2) Department of Mathematics, Texas A\& M University, College Station TX, 77843-3368, U.S.A.

Summary: A finite von Neumann algebra $\mathcal{M}$ with a faithful normal trace $ \tau $ has Haagerup's approximation property if there exists a pointwise deformation of the identity in $2$-norm by subtracial compact completely positive maps. In this paper we prove that the subtraciality condition can be removed. This enables us to provide a description of Haagerup's approximation property in terms of correspondences. We also show that if $ \mathcal{N}\subset \mathcal{M}$ is an amenable inclusion of finite von Neumann algebras and $\mathcal{N}$ has Haagerup's approximation property, then $\mathcal{M}$ also has Haagerup's approximation property.

Keywords: Von Neumann algebras, Haagerup's approximation property, relative Haagerup's approximation property, relative amenability

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