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Amenable dynamical systems over locally compact groups

Part of: Selfadjoint operator algebras Linear spaces and algebras of operators

Published online by Cambridge University Press:  25 June 2021

ALEX BEARDEN  and
JASON CRANN Open the ORCID record for JASON CRANN [Opens in a new window]
ALEX BEARDEN
Affiliation:
Department of Mathematics, University of Texas at Tyler, Tyler, TX 75799, USA (e-mail: cbearden@uttyler.edu)
JASON CRANN*
Affiliation:
School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6
*
e-mail: jasoncrann@cunet.carleton.ca
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Abstract

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 $(M,G,\alpha )$ converging point weak* to the identity of $G\bar {\ltimes }M$. In the $C^*$-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 $G\ltimes A$, as well as a particular case of the positive weak approximation property of Bédos and Conti [On discrete twisted $C^*$-dynamical systems, Hilbert $C^*$-modules and regularity. Münster J. Math. 5 (2012), 183–208] (generalized to 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 [Approximation property of $C^*$-algebraic bundles. Math. Proc. Cambridge Philos. Soc. 132(3) (2002), 509–522]. In particular, when $A=C_0(X)$ is commutative, amenability of $(C_0(X),G,\alpha )$ coincides with topological amenability of the G-space $(G,X)$.

Keywords

dynamical systemscrossed productslocally compact groupsamenable actions

MSC classification

Primary: 46L55: Noncommutative dynamical systems
Secondary: 47L65: Crossed product algebras (analytic crossed products) 46L07: Operator spaces and completely bounded maps

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 8 , August 2022 , pp. 2468 - 2508
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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