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On graph products of multipliers and the Haagerup property for $C^{\ast }$-dynamical systems

Part of: Abstract harmonic analysis Selfadjoint operator algebras Locally compact groups and their algebras Special aspects of infinite or finite groups

Published online by Cambridge University Press:  07 August 2019

SCOTT ATKINSON
SCOTT ATKINSON*
Affiliation:
Vanderbilt University, Department of Mathematics, 1326 Stevenson Center, Station B 407807, Nashville, TN37240, USA email scott.a.atkinson@vanderbilt.edu
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Abstract

We consider the notion of the graph product of actions of discrete groups $\{G_{v}\}$ on a $C^{\ast }$-algebra ${\mathcal{A}}$ and show that under suitable commutativity conditions the graph product action $\star _{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D6FC}_{v}:\star _{\unicode[STIX]{x1D6E4}}G_{v}\curvearrowright {\mathcal{A}}$ has the Haagerup property if each action $\unicode[STIX]{x1D6FC}_{v}:G_{v}\curvearrowright {\mathcal{A}}$ possesses the Haagerup property. This generalizes the known results on graph products of groups with the Haagerup property. To accomplish this, we introduce the graph product of multipliers associated to the actions and show that the graph product of positive-definite multipliers is positive definite. These results have impacts on left-transformation groupoids and give an alternative proof of a known result for coarse embeddability. We also record a cohomological characterization of the Haagerup property for group actions.

Keywords

operator algebrasgroup actionsC∗ -dynamical systemsHaagerup propertypositive-definite multipliers

MSC classification

Primary: 46L55: Noncommutative dynamical systems
Secondary: 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations 43A35: Positive definite functions on groups, semigroups, etc. 20F65: Geometric group theory
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 12 , December 2020 , pp. 3188 - 3216
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Copyright
© Cambridge University Press, 2019

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