Haagerup's Approximation Property and Relative Amenability

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 $(\phi_{\alpha}){\alpha\in \Lambda}$ofnormalcompletelypositive($$\mathcal{N}$$-bimodular)mapsfrom$$\mathcal{M}$$to$$\mathcal{M}$$thatsatisfythesubtracialcondition$% \tau \circ \phi{\alpha}\leq \tau $,theextensionoperators$% T_{\phi_{\alpha}}$areboundedcompactoperators(in$<\mathcal{M%},e_{$\mathcal{N}$}>$), and pointwise approximate the identity in the trace-norm, i.e., ${\lim }_{\alpha }\mathrm{\mid }\mathrm{\mid }{\varphi }_{\alpha }(x)-x\mathrm{\mid }{\mathrm{\mid }}_2=0$ 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.