Haagerup's Approximation Property and Relative Amenability

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