Haagerup's Approximation Property and Relative Amenability

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.