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The Unitary Conjugation Groupoid of a Type I C*-Algebra: Topology, Fell Continuity, and the Canonical Diagonal Embedding

Shih-Yu Chang

Abstract

This paper introduces a canonical Polish groupoid associated to any separable unital C*-algebra, termed the unitary conjugation groupoid. It is defined as the semidirect product of the algebra's dual space by its unitary group, acting by conjugation. Classical groupoid models for C*-algebras typically require additional structure such as a Cartan subalgebra and rely on the locally compact Hausdorff framework. In contrast, our construction is entirely canonical but forces a paradigm shift: the natural topologies on the dual space and the unitary group are not locally compact. To address this, we equip the dual space with a Polish topology derived from the weak-star topology on pure states and the unitary group with the strong operator topology. This yields a Polish groupoid admitting a continuous Haar system. We prove that the associated reduced groupoid C*-algebra is Morita equivalent to the original algebra tensored with the compact operators, establishing that the groupoid encodes the K-theory of the algebra. A key feature is a canonical diagonal embedding of the original algebra into the groupoid C*-algebra, which is a unital injective star-homomorphism for separable Type I C*-algebras. We characterize commutativity via this embedding and establish functoriality of the construction under appropriate star-homomorphisms. The theory is illustrated with detailed computations for three fundamental classes: finite-dimensional matrix algebras, commutative algebras over compact metrizable spaces, and the unitized compact operators. These examples demonstrate that our general constructions reduce to familiar objects. We also discuss limitations by analyzing the irrational rotation algebra, a non-Type I algebra not covered by our construction, highlighting directions for future research.

The Unitary Conjugation Groupoid of a Type I C*-Algebra: Topology, Fell Continuity, and the Canonical Diagonal Embedding

Abstract

This paper introduces a canonical Polish groupoid associated to any separable unital C*-algebra, termed the unitary conjugation groupoid. It is defined as the semidirect product of the algebra's dual space by its unitary group, acting by conjugation. Classical groupoid models for C*-algebras typically require additional structure such as a Cartan subalgebra and rely on the locally compact Hausdorff framework. In contrast, our construction is entirely canonical but forces a paradigm shift: the natural topologies on the dual space and the unitary group are not locally compact. To address this, we equip the dual space with a Polish topology derived from the weak-star topology on pure states and the unitary group with the strong operator topology. This yields a Polish groupoid admitting a continuous Haar system. We prove that the associated reduced groupoid C*-algebra is Morita equivalent to the original algebra tensored with the compact operators, establishing that the groupoid encodes the K-theory of the algebra. A key feature is a canonical diagonal embedding of the original algebra into the groupoid C*-algebra, which is a unital injective star-homomorphism for separable Type I C*-algebras. We characterize commutativity via this embedding and establish functoriality of the construction under appropriate star-homomorphisms. The theory is illustrated with detailed computations for three fundamental classes: finite-dimensional matrix algebras, commutative algebras over compact metrizable spaces, and the unitized compact operators. These examples demonstrate that our general constructions reduce to familiar objects. We also discuss limitations by analyzing the irrational rotation algebra, a non-Type I algebra not covered by our construction, highlighting directions for future research.
Paper Structure (45 sections, 139 theorems, 267 equations)

This paper contains 45 sections, 139 theorems, 267 equations.

Table of Contents

  1. Introduction
  2. Context: Classical Perspectives on Noncommutative Algebras
  3. The Unitary Conjugation Groupoid: A New Invariant
  4. Main Obstacles: Étaleness and Local Compactness are Unattainable
  5. A Paradigm Shift: From Local Compactness to Polish Topology
  6. Main Contributions and Outline of the Paper
  7. Organization of the Paper
  8. Preliminaries
  9. Type I C*-Algebras and their Primitive Ideal Spaces
  10. The Fell Topology on the Space of Subalgebras
  11. Polish Spaces and the Strong Operator Topology
  12. Polish Groupoids and their C*-Algebras
  13. The Unit Space $\mathcal{G}_{\mathcal{A}}^{(0)}$: Definition and Polish Topology
  14. Definition of $\mathcal{G}_{\mathcal{A}}^{(0)}$ and Partial Evaluation Maps
  15. Initial Topology and Basic Properties
  16. ...and 30 more sections

Key Result

Theorem 1

Let $\mathcal{A}$ be a separable C*-algebra. Then $\mathcal{A}$ is Type I if and only if its primitive ideal space $\operatorname{Prim}(\mathcal{A})$ is almost Hausdorff; that is, every nonempty closed subset of $\operatorname{Prim}(\mathcal{A})$ contains a nonempty relatively open Hausdorff subset.

Theorems & Definitions (470)

  • Remark 1
  • Definition 1: Type I C*-algebra
  • Definition 2
  • Theorem 1: Glimm--Dixmier Theorem
  • Definition 3: Primitive ideal space
  • Remark 2
  • Remark 3: Type I examples
  • Example 1: Finite-dimensional algebras
  • Example 2: Commutative algebras
  • Example 3: Compact operators
  • ...and 460 more