We establish a Reiter property for amenable W * -dynamical systems (M, G, α) over arbitrary locally compact groups. We also prove that a commutative C * -dynamical system (C 0 (X), G, α) is topologically amenable if and only if its universal W * -dynamical system is amenable. Our results answer three open questions from the literature; one of Anantharaman-Delaroche from [3], and two from a recent preprint of Buss-Echterhoff-Willett [9].

We introduce and study a generalization of the notion of the Furstenberg boundary of a discrete group $\Gamma $ to the setting of a general unitary representation $\pi : \Gamma \to B(\mathcal H_\pi )$. This space, which we call the “Furstenberg–Hamana boundary” (or "FH-boundary") of the pair $(\Gamma , \pi )$, is a $\Gamma $-invariant subspace of $B(\mathcal H_\pi )$ that carries a canonical $C^{\ast }$-algebra structure. In many natural cases, including when $\pi $ is a quasi-regular representation, the Furstenberg–Hamana boundary of $\pi $ is commutative but can be noncommutative in general. We study various properties of this boundary and discuss possible applications, for example in uniqueness of certain types of traces.

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.

We introduce and study a generalization of the notion of the Furstenberg boundary of a discrete group Γ to the setting of a general unitary representation π : Γ → B(H π ). This space, which we call the "Furstenberg-Hamana boundary" of the pair (Γ, π) is a Γ-invariant subspace of B(H π ) that carries a canonical C * -algebra structure. In many natural cases, including when π is a quasiregular representation, the Furstenberg-Hamana boundary of π is commutative, but can be non-commutative in general. We study various properties of this boundary, and give some applications.

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 [8]. 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, α) with M injective is amenable if and only if M is L 1 (G)-injective if and only if the crossed product G ⋉M is A(G)-injective. Analogously, we show that a C * -dynamical system (A, G, α) 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 ⋉ A has the A(G)-WEP.