TITLE: Characterization of amenable actions
QUESTION [7 upvotes]: Let $(X,\mu)$ be a $G$-space, i.e. a measure space with a Borel quasi-invariant $G$-action. Say that $X$ is amenable (equivalently, that the action is amenable) if there is a $G$-fixed point in every affine space over $X$ with an $\alpha$-twisted action, where $\alpha$ is a corresponding cocycle.
If $X$ is an amenable $G$-space, it follows more or less from definition that for every compact metric space $Y$ there is an  $G$-equivariant measurable map $\varphi : X \to M(Y)$, where $M(Y)$ is the collection of probability measures on $Y$.
My question is - is this property equivalent to the above definition of amenable action or strictly weaker?

REPLY [3 votes]: We have resolved this question recently in Proposition 3.16 of
https://arxiv.org/pdf/2003.03469.pdf
We prove, among other things, that an action of an exact locally compact group G on a von Neumann algebra M is amenable (in the sense of Claire Anantharaman-Delaroche, i.e., there is a $G$-equivariant projection $M\bar\otimes L^\infty(G)\to M$) if and only if there is a unital G-equivariant ucp map $L^\infty(G)\to Z(M)$, the center of $M$, or equivalently, a unital $G$-equivariant ucp map $C_{ub}(G)\to Z(M)$, where $C_{ub}(G)=UC_b(G)$ denotes the algebra of bounded uniformly continuous functions $G\to \mathbb{C}$ as in Jesse Peterson's answer.
The existence of $C_{ub}(G)\to Z(M)$ is equivalent to the existence of a $G$-equivariant projection $Z(M)\otimes C_{ub}(G)\to Z(M)$; one can also replace $C_{ub}(G)$ by $L^\infty(G)$ here. But $\otimes$ here denotes the $C^*$-tensor product, and it cannot be replaced by the von Neumann tensor product $\bar\otimes$ in general, even if $G$ is discrete.
Indeed, if $G$ is not exact, then $A=\ell^\infty(G)$ is a commutative $G$-$C^*$-algebra with respect to the translation $G$-action. And this action (or equivalently the $G$-action on its spectrum $\beta G$) is amenable if and only if $G$ is exact. This means that the $G$-action on the bidual von Neumann algebra $A^{**}$ is not von Neumann amenable (in the sense of Delaroche).