Amenable dynamical systems over locally compact groups

We establish several new characterizations of amenable $W^*$- and $C^*$-dynamical systems over arbitrary locally compact groups. In the $W^*$-setting we show that amenability is equivalent to (1) a Reiter property and (2) the existence of a certain net of completely positive Herz-Schur multipliers of $(M,G,α)$ converging point weak* to the identity of $G\bar{\ltimes}M$. In the $C^*$-setting, we prove that amenability of $(A,G,α)$ is equivalent to an analogous Herz-Schur multiplier approximation of the identity of the reduced crossed product $G\ltimes A$, as well as a particular case of the positive weak approximation property of Bédos and Conti (generalized the locally compact setting). When $Z(A^{**})=Z(A)^{**}$, it follows that amenability is equivalent to the 1-positive approximation property of Exel and Ng. In particular, when $A=C_0(X)$ is commutative, amenability of $(C_0(X),G,α)$ coincides with topological amenability the $G$-space $(G,X)$. Our results answer 2 open questions from the literature; one of Anantharaman--Delaroche, and one from recent work of Buss--Echterhoff--Willett.

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