Amenable dynamical systems over locally compact groups

Abstract

We establish several new characterizations of amenable $W^{\ast }$- and $C^{\ast }$-dynamical systems over arbitrary locally compact groups. In the $W^{\ast }$-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 $(M,G,\alpha )$ converging point weak* to the identity of $G\overline{⋉}M$. In the $C^{\ast }$-setting, we prove that amenability of $(A,G,\alpha )$ is equivalent to an analogous Herz-Schur multiplier approximation of the identity of the reduced crossed product $G⋉A$, as well as a particular case of the positive weak approximation property of Bédos and Conti (generalized the locally compact setting). When $Z(A^{\ast \ast })=Z(A{)}^{\ast \ast }$, 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 $(C_0(X),G,\alpha )$ 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.

v2: Substantial changes. Corrected an error in the proof of (the old) Theorem 4.2. Added new results concerning Herz-Schur multipliers and the weak approximation property. New length is 36 pages