A weak expectation property for operator modules, injectivity and amenable actions

We introduce an equivariant version of the weak expectation property (WEP) at the level of operator modules over completely contractive Banach algebras $A$. We prove a number of general results---for example, a characterization of the $A$-WEP in terms of an appropriate $A$-injective envelope, and also a characterization of those $A$ for which $A$-WEP implies WEP. In the case of $A=L^1(G)$, we recover the $G$-WEP for $G$-$C^*$-algebras in recent work of Buss--Echterhoff--Willett. When $A=A(G)$, we obtain a dual notion for operator modules over the Fourier algebra. These dual notions are related in the setting of dynamical systems, where we show that a $W^*$-dynamical system $(M,G,\alpha)$ with $M$ injective is amenable if and only if $M$ is $L^1(G)$-injective if and only if the crossed product $G\bar{\ltimes}M$ is $A(G)$-injective. Analogously, we show that a $C^*$-dynamical system $(A,G,\alpha)$ with $A$ nuclear and $G$ exact is amenable if and only if $A$ has the $L^1(G)$-WEP if and only if the reduced crossed product $G\ltimes A$ has the $A(G)$-WEP.