Haagerup's Approximation Property and Relative Amenability

TL;DR

This paper refines the understanding of Haagerup's approximation property in finite von Neumann algebras, removing certain conditions, and demonstrates how it propagates through amenable inclusions, answering key questions in the field.

Contribution

It removes the subtraciality condition from the definition of Haagerup's property and links it to Connes's correspondence theory, also showing inheritance under amenable subalgebra inclusions.

Findings

Subtraciality condition can be removed from the definition.

Haagerup's property can be characterized via Connes's correspondences.

Haagerup's property passes from an amenable subalgebra to the larger algebra.

Abstract

A finite von Neumann algebra $\mathcal{M}$ with a faithful normal trace $% \tau $ has Haagerup's approximation property (relative to a von Neumann subalgebra $\mathcal{N}$) if there exists a net $(\phi_{\alpha})_{\alpha\in \Lambda}$ of normal completely positive ($\mathcal{N}$-bimodular) maps from $\mathcal{M}$ to $\mathcal{M}$ that satisfy the subtracial condition $% \tau \circ \phi_{\alpha}\leq \tau $, the extension operators $% T_{\phi_{\alpha}}$ are bounded compact operators (in $<\mathcal{M%},e_{\mathcal{N}}>$), and pointwise approximate the identity in the trace-norm, i.e., $\lim_{\alpha}||\phi_{\alpha}(x)-x||_{2}=0$ for all $% x\in \mathcal{M}$. We prove that the subtraciality condition can be removed, and provide a description of Haagerup's approximation property in terms of Connes's theory of correspondences. We show that if $\mathcal{N}\subseteq \mathcal{M}$ 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. This work answers two questions of Sorin Popa.