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Let $\cal{M}$ be a finite von Neumann algebra and $\cal{N}$ be a von Neumann subalgebra of $\cal{M}$. The von Neumann algebra $\cal{M}$ is is amenable relative to $\cal{N}$ if there exists a norm one projection of $<\cal{M},\cal{N}>$ onto $\cal{M}$. See Nicolas Monod, Sorin Popa, On co-Amenability for groups and von Neumann algebras. (https://arxiv.org/abs/math/0301348).

Question: Can we characterize the relative amenability of von Neumann algebras in terms of first Hochschild cohomology group?

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    $\begingroup$ Have you considered the case $\mathcal N = \mathbb C$? $\endgroup$
    – Jesse Peterson
    Commented Nov 3, 2017 at 2:13
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    $\begingroup$ Have you taken a look at arxiv.org/abs/1511.07329 ? $\endgroup$
    – Marcel Bischoff
    Commented Jan 11, 2018 at 19:23

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