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.
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