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Principal Actions on Topological Quivers and Associated Operator Dynamics

Matthew Gillespie, Lucas Hall, Benjamin Jones, Mariusz Tobolski

TL;DR

The paper develops a complete classification of principal, free, and proper actions of a locally compact group $G$ on topological quivers and proves that the quiver algebra of the quotient $C^*(Q/G)$ is isomorphic to the fixed-point algebra $C^*(Q)^α$, which is Morita equivalent to the reduced crossed product $C^*(Q)\rtimes_r G$. Using functoriality for $C^*$-correspondences and CP covariance, the authors extend the Deaconu--Kumjian--Quigg program to the topological-quiver setting, introducing edge-measure systems to handle non-discrete vertex spaces. They illustrate the theory with many concrete coset- and skew-product constructions, semidirect-product cosets, non-split extensions, and topological-group-relations, highlighting new Morita equivalences beyond topological graphs. The results offer a flexible framework for studying noncommutative spaces arising from group actions on generalized graphs and connect to nonabelian duality and Rieffel’s fixed-point theory, broadening the landscape of operator-algebraic dynamics on topological quivers.

Abstract

We study topological quivers $Q$ admitting a free and proper action by a locally compact group $G$ together with their associated $C^*$-algebras. On the topological side, we provide a complete classification of topological quivers which admit such actions in terms of $G$-bundles over the vertex orbit space and an appropriate isomorphism of bundles over the edge orbits. Following the work by Deaconu, Kumjian, and Quigg on topological graphs, we construct an isomorphism between $C^*(Q/G)$ and Rieffel's fixed-point algebra $C^*(Q)^α$, which is known to be Morita equivalent to $C^*(Q)\rtimes_rG$. Unlike the work with topological graphs, we use previously developed functoriality techniques to identify the isomorphism. We also examine many concrete examples of such group actions, including some exclusive to topological quivers, and the associated Morita equivalences.

Principal Actions on Topological Quivers and Associated Operator Dynamics

TL;DR

The paper develops a complete classification of principal, free, and proper actions of a locally compact group on topological quivers and proves that the quiver algebra of the quotient is isomorphic to the fixed-point algebra , which is Morita equivalent to the reduced crossed product . Using functoriality for -correspondences and CP covariance, the authors extend the Deaconu--Kumjian--Quigg program to the topological-quiver setting, introducing edge-measure systems to handle non-discrete vertex spaces. They illustrate the theory with many concrete coset- and skew-product constructions, semidirect-product cosets, non-split extensions, and topological-group-relations, highlighting new Morita equivalences beyond topological graphs. The results offer a flexible framework for studying noncommutative spaces arising from group actions on generalized graphs and connect to nonabelian duality and Rieffel’s fixed-point theory, broadening the landscape of operator-algebraic dynamics on topological quivers.

Abstract

We study topological quivers admitting a free and proper action by a locally compact group together with their associated -algebras. On the topological side, we provide a complete classification of topological quivers which admit such actions in terms of -bundles over the vertex orbit space and an appropriate isomorphism of bundles over the edge orbits. Following the work by Deaconu, Kumjian, and Quigg on topological graphs, we construct an isomorphism between and Rieffel's fixed-point algebra , which is known to be Morita equivalent to . Unlike the work with topological graphs, we use previously developed functoriality techniques to identify the isomorphism. We also examine many concrete examples of such group actions, including some exclusive to topological quivers, and the associated Morita equivalences.

Paper Structure

This paper contains 12 sections, 12 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Quivers and Bundles
  5. Operator Algebras
  6. Actions on Quivers and Examples
  7. Topological quivers coming from coset spaces
  8. Coset quivers of semi-direct products and skew-products
  9. Non-split extensions of groups
  10. Topological group relations
  11. Topological Aspects
  12. Correspondences

Key Result

Theorem 2.2

Suppose we are given principal $G$-bundles $q_{i}:P_i\rightarrow X_i\cong P_{i}/G$ for $i=1,2$ and a pair of continuous maps $f:X_{1}\rightarrow X_{2}$, $\widetilde{f}:P_{1}\rightarrow P_{2}$ such that $\widetilde{f}$ is $G$-equivariant and $q_{2}\circ\widetilde{f}=f\circ q_{1}$. Then $\theta_{f}:P_

Theorems & Definitions (44)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3: kqrfunctor*Definition 3.1
  • Definition 3.1
  • Definition 3.2
  • Example 3.3
  • Definition 3.4
  • Example 3.5: Toeplitz algebra
  • Example 3.6: Irrational rotation algebra
  • Example 3.7: Cuntz algebras
  • ...and 34 more