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RFD property for groupoid C*-algebras of amenable groupoids and for crossed products by amenable actions

Tatiana Shulman, Adam Skalski

Abstract

By Bekka's theorem the group C*-algebra of an amenable group $G$ is residually finite dimensional (RFD) if and only if $G$ is maximally almost periodic (MAP). We generalize this result in two directions of dynamical flavour. Firstly, we provide a sufficient condition for the RFD property of the C*-algebra of an amenable étale groupoid. Secondly, we characterize RFD property for crossed products by amenable actions of discrete groups on C*-algebras. The characterisation can be formulated in various terms, such as primitive ideals, (pure) states and approximations of representations, and can be viewed as a dynamical version of Exel-Loring characterization of RFD C*-algebras. As byproduct of our methods we also characterize the property FD of Lubotzky and Shalom for semidirect products by amenable groups and obtain characterizations of the properties MAP and RF for general semidirect products of groups. The latter descriptions allow us to obtain the properties MAP, RF, RFD and FD for various examples.

RFD property for groupoid C*-algebras of amenable groupoids and for crossed products by amenable actions

Abstract

By Bekka's theorem the group C*-algebra of an amenable group is residually finite dimensional (RFD) if and only if is maximally almost periodic (MAP). We generalize this result in two directions of dynamical flavour. Firstly, we provide a sufficient condition for the RFD property of the C*-algebra of an amenable étale groupoid. Secondly, we characterize RFD property for crossed products by amenable actions of discrete groups on C*-algebras. The characterisation can be formulated in various terms, such as primitive ideals, (pure) states and approximations of representations, and can be viewed as a dynamical version of Exel-Loring characterization of RFD C*-algebras. As byproduct of our methods we also characterize the property FD of Lubotzky and Shalom for semidirect products by amenable groups and obtain characterizations of the properties MAP and RF for general semidirect products of groups. The latter descriptions allow us to obtain the properties MAP, RF, RFD and FD for various examples.
Paper Structure (23 sections, 56 theorems, 118 equations)

This paper contains 23 sections, 56 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. RFD Property for groupoid $C^*$-algebras
  3. RFD property for groupoid $C^*$-algebras -- positive results
  4. Example of RFD amenable groupoid with a non-MAP isotropy subgroup
  5. A characterization of RFD crossed products
  6. Approximations via periodic finite-dimensional representations
  7. Finite-dimensional (and not only) representations of $A \rtimes G$
  8. Characterisation of the RFD property of crossed products
  9. Some consequences
  10. Automatic amenability
  11. Non-amenable case
  12. Semi-direct products of groups
  13. RFD and MAP
  14. FD and RF
  15. Examples
  16. ...and 8 more sections

Key Result

Theorem 1

Let $G$ be a discrete group, $A$ a C*-algebra and $\alpha$ an amenable action of $G$ on $A$. Then the following conditions are equivalent:

Theorems & Definitions (118)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 108 more