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

Abstract

We introduce an equivariant version of the weak expectation property (WEP) at the level of operator modules over completely contractive Banach algebras [Formula: see text]. We prove a number of general results — for example, a characterization of the [Formula: see text]-WEP in terms of an appropriate [Formula: see text]-injective envelope, and also a characterization of those [Formula: see text] for which [Formula: see text]-WEP implies WEP. In the case of [Formula: see text], we recover the [Formula: see text]-WEP for [Formula: see text]-[Formula: see text]-algebras in recent work of Buss–Echterhoff–Willett [A. Buss, S. Echterhoff and R. Willett, The maximal injective crossed product, to appear in Ergodic Theory Dynam. Systems, https://doi.org/10.1017/etds.2019.25 ]. When [Formula: see text], 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 [Formula: see text]-dynamical system [Formula: see text] with [Formula: see text] injective is amenable if and only if [Formula: see text] is [Formula: see text]-injective if and only if the crossed product [Formula: see text] is [Formula: see text]-injective. Analogously, we show that a [Formula: see text]-dynamical system [Formula: see text] with [Formula: see text] nuclear and [Formula: see text] exact is amenable if and only if [Formula: see text] has the [Formula: see text]-WEP if and only if the reduced crossed product [Formula: see text] has the [Formula: see text]-WEP.