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

TL;DR

This paper introduces an equivariant weak expectation property for operator modules over Banach algebras, characterizes its relation to injectivity and amenability, and connects these notions in dynamical systems and crossed products.

Contribution

It develops a new equivariant WEP for operator modules, characterizes it via injective envelopes, and links it to amenability in dynamical systems and crossed product constructions.

Findings

A characterization of the $A$-WEP in terms of $A$-injective envelopes.

Equivalence of $A$-WEP and WEP for certain algebras.

Connection between injectivity, WEP, and amenability in dynamical systems.

Abstract

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.