Characterization of amenable actions 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?
 A: Here is a sketch of an argument to show these properties are equivalent which is based on the proof of the same property for amenable groups. To see this I will use the fact that an action $G \curvearrowright (X, \mu)$ is amenable if and only if there is a $G$-equivariant mean from $L^\infty(X, \mu) \overline \otimes L^\infty(G)$ to $L^\infty(X, \mu)$. (A mean in this setting is a positivity preserving map which restricts to the identity on $L^\infty(X, \mu)$.) This is Theorem A (iv) in "Amenable Actions of Groups" by Adams, Elliott, Giordano (1994) (http://www.ams.org/mathscinet-getitem?mr=1250814), which generalizes the case for countable groups established by Proposition 4.1 in "Hyperfinite Factors and Amenable Ergodic Actions" by Zimmer (1977) (http://www.ams.org/mathscinet-getitem?mr=470692).
Theorem: Let $G$ be a locally compact second countable group. Then there exists a $G$-equivariant mean from $L^\infty(X, \mu) \overline \otimes L^\infty(G)$ to $L^\infty(X, \mu)$ if and only if for every continuous action of $G$ on a compact metric space $Y$, there exists a $G$-equivariant map from $X$ to $M(Y)$.
Proof: For the proof I will find it easier to work with function spaces, and so first note that for a compact Hausdorff space $K$, there is a one-to-one correspondence between (equivalence classes of) Borel maps $\pi: X \to M(K)$ and means $\Phi: L^\infty(X, \mu) \otimes C(K) \to L^\infty(X, \mu)$. (Here, $\otimes$ is the $C^*$-tensor product.) This is essentially just the Riesz representation theorem applied to each fiber $K$.
We now show the "only if" direction. If $G \curvearrowright K$ is a continuous action on a compact metric space, by restricting to a closed $G$-invariant subset we may assume that $G \curvearrowright K$ has a dense orbit $K = \overline{G k}$. We then obtain a $G$-equivariant embedding $\rho: C(K) \to C_b(G) \subset L^\infty(G)$ by $\rho(f)(g) = f(gk)$. Restricting our invariant mean to $L^\infty(X, \mu) \otimes \rho(C(K))$ then gives the result.
The converse is a bit more involved, but this is well known for the case of amenable groups. The first step is to show that we have an equivariant mean on the space $UC_b(G)$ of bounded left uniformly continuous functions. In the case when $G$ is countable we are then done. For the general case we then use an approximate identity for  convolution to produce an equivariant mean for $L^\infty(G)$.
To produce a mean on the space $UC_b(G)$ note that for any second countable $G$-invariant $C^*$-subalgebra $A \subset UC_b(G)$ we have that the Gelfand spectrum is metrizable, and $G$ acts continuously. Therefore by hypothesis there exists a $G$-equivariant mean $\Phi_A: L^\infty(X, \mu) \otimes A \to L^\infty(X, \mu)$. Since $L^\infty(X, \mu)$ is injective we may then extend $\Phi_A$ to a (perhaps no longer $G$-equivariant) positivity preserving map $\widetilde{\Phi_A} : L^\infty(X, \mu) \otimes UC_b(G) \to L^\infty(X, \mu)$. If we consider the net $\{ \widetilde{\Phi_A} \}$ indexed by the set of all second countable $G$-invariant $C^*$-subalgebras $A$, and ordered by inclusion, then we may take an accumulation point $\Phi$ in the topology of point-wise weak convergence. Since $\Phi$ is $G$-equivariant when restricted to $L^\infty(X, \mu) \otimes A$ for any second countable $C^*$-subalgebra $A$, it follows that $\Phi$ is $G$-equivariant on the whole space.
To produce a mean on $L^\infty(X, \mu) \overline \otimes L^\infty(G)$ we start by taking an approximate identity $\{ \psi_n \} \subset C_c(G)$. Specifically, we want that each $\psi_n \in C_c(G)$ is a non-negative function, $\| \psi_n \|_1 = 1$, ${\rm supp}(\psi_n) \to \{ e \}$, and $\| \psi_n * \delta_g - \delta_g * \psi_n \|_1 \to 0$ for each $g \in G$. If $f \in L^\infty(X, \mu) \overline \otimes L^\infty(G)$, then we have $\psi_n * f \in L^\infty(X, \mu) \otimes UC_b(G)$ for each $n$ (I'm taking convolution pointwise in $X$), and $\|  \sigma_g(\psi_n * f) - \psi_n * (\sigma_g(f)) \|_\infty \to 0$ for each $g \in G$, and $f \in L^\infty(X, \mu) \overline \otimes L^\infty(G)$. (We denote by $\sigma_g$ the action of $G$ on $L^\infty(X, \mu) \overline \otimes L^\infty(G)$.) If we set $\Phi_n : L^\infty(X, \mu) \overline \otimes L^\infty(G) \to L^\infty(X, \mu)$, by $\Phi_n( f ) = \Phi( \psi_n * f )$, then it follows that any accumulation point of $\{ \Phi_n \}$ gives us a $G$-equivariant mean. $\blacksquare$
Update:Nicolas Monod pointed out to me that, in my sketch above, the way that I use the approximate identity $\{ \psi_n \}$ at the end is not correct. So the proof that I outline seems to work just fine for discrete groups, but the general case for locally compact groups appears more difficult. In fact, it appears to be related to the notion of "relative amenability" presented in:
Caprace, Pierre-Emmanuel; Monod, Nicolas: Relative amenability. Groups Geom. Dyn. 8 (2014), no. 3, 747–774.
Update 2: Even with discrete groups this answer only works with the $C^*$-tensor product. However, the answer by Buss points to their paper containing a very satisfying resolution of this issue through the use of exactness.
A: 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).
