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Weyl groups of groupoid C*-algebras

Fuyuta Komura

TL;DR

The paper develops Weyl and restricted Weyl groups for groupoid C*-algebras, providing a unifying framework that generalizes the Weyl groups known for Cuntz and graph algebras. It proves that for effective groupoids, Aut$(C^*_r(G);C_0(G^{(0)}))$ is isomorphic to Aut$(G) times Z(G)$, with Aut$_{C_0(G^{(0)})}(C^*_r(G)) vert_{Z(G)}$ identifying the $Z(G)$-part; in particular, the Weyl group coincides with Aut$(G)$ when $G$ is effective, and is discrete/countable when $G$ is expansive. The work introduces restricted Weyl groups for inclusions $G^{(0)} o H o G$, linking them to cocycles via $ ext{ker}\sigma$ and to automorphisms compatible with the cocycle; it further connects compact abelian actions to discrete cocycles and identifies fixed point algebras with groupoid C*-algebras of kernel groupoids, yielding Galois-dual descriptions. Applications to Deaconu–Renault systems and graph algebras include a resolution of an open problem on graph restricted Weyl groups and a new result for the infinite-degree Cuntz algebra, illustrating the power of the groupoid approach to intertwine operator algebras with symbolic dynamics.

Abstract

In the theory of C*-algebras, the Weyl groups were defined for the Cuntz algebras and graph algebras by Cuntz and Conti et al. respectively. In this paper, we introduce and investigate the Weyl groups of groupoid C*-algebras as a natural generalization of the existing Weyl groups. Then we analyse several groups of automorphisms on groupoid C*-algebras. Finally, we apply our results to Cuntz algebras, graph algebras and C*-algebras associated with Deaconu-Renault systems.

Weyl groups of groupoid C*-algebras

TL;DR

The paper develops Weyl and restricted Weyl groups for groupoid C*-algebras, providing a unifying framework that generalizes the Weyl groups known for Cuntz and graph algebras. It proves that for effective groupoids, Aut is isomorphic to Aut, with Aut identifying the -part; in particular, the Weyl group coincides with Aut when is effective, and is discrete/countable when is expansive. The work introduces restricted Weyl groups for inclusions , linking them to cocycles via and to automorphisms compatible with the cocycle; it further connects compact abelian actions to discrete cocycles and identifies fixed point algebras with groupoid C*-algebras of kernel groupoids, yielding Galois-dual descriptions. Applications to Deaconu–Renault systems and graph algebras include a resolution of an open problem on graph restricted Weyl groups and a new result for the infinite-degree Cuntz algebra, illustrating the power of the groupoid approach to intertwine operator algebras with symbolic dynamics.

Abstract

In the theory of C*-algebras, the Weyl groups were defined for the Cuntz algebras and graph algebras by Cuntz and Conti et al. respectively. In this paper, we introduce and investigate the Weyl groups of groupoid C*-algebras as a natural generalization of the existing Weyl groups. Then we analyse several groups of automorphisms on groupoid C*-algebras. Finally, we apply our results to Cuntz algebras, graph algebras and C*-algebras associated with Deaconu-Renault systems.
Paper Structure (21 sections, 54 theorems, 143 equations)

This paper contains 21 sections, 54 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Étale groupoids
  4. Inverse semigroup actions
  5. Groupoid C*-algebras
  6. Automorphism groups
  7. Miscellaneous facts
  8. Weyl groups of groupoid C*-algebras
  9. Basic properties of Weyl groups
  10. Restricted Weyl groups of groupoid C*-algebras
  11. Restricted Weyl group (general case)
  12. Restricted Weyl group (discrete group cocycle kernel case)
  13. Compact abelian group actions
  14. Examples and applications
  15. Examples of coactions on the Cuntz algebras
  16. ...and 6 more sections

Key Result

Proposition 1.3.1

Let $G$ be a locally compact Hausdorff étale groupoid. For $a\in C^*_r(G)$, $j(a)\in C_0(G)$ is defined by for $\alpha\in G$In this paper, inner products of Hilbert spaces are linear with respect to the right variables.. Then $j\colon C^*_r(G)\to C_0(G)$ is a norm decreasing injective linear map. Moreover, $j$ is an identity map on $C_c(G)$.

Theorems & Definitions (88)

  • Example 1.2.1: paterson2012groupoids
  • Proposition 1.3.1: renault1980groupoid
  • Remark 1.3.2
  • Definition 1.4.1
  • Remark 1.4.2
  • Proposition 1.4.3
  • Proposition 1.5.1
  • Proposition 1.5.2
  • Proposition 1.5.3
  • Remark 1.5.4
  • ...and 78 more