Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The primitive spectrum of C*-algebras of etale groupoids with abelian isotropy

Johannes Christensen, Sergey Neshveyev

TL;DR

This work provides a topological description of the primitive spectrum $\operatorname{Prim} C^*(\mathcal{G})$ for amenable, second countable, étale groupoids with abelian isotropy by identifying it with a quotient of the character space $\operatorname{Char}(\mathcal{G})$ via induction from isotropy. The authors introduce the spaces $\operatorname{Char}(\mathcal{G})$ and $\operatorname{Stab}(\mathcal{G})\widehat{\ }$, establish a robust induction map, and prove that, under the stated hypotheses, $\operatorname{Ind}^{\sim}$ yields a homeomorphism from $(\mathcal{G}\backslash\operatorname{Stab}(\mathcal{G})\widehat{\ })^{\sim}$ onto $\operatorname{Prim} C^*(\mathcal{G})$. They develop (i) a general framework for injectively graded groupoids, (ii) concrete results for transformation groupoids, Deaconu–Renault groupoids, and (iii) explicit descriptions for graph algebras, including 1-graphs and higher-rank graphs. Consequently, the ideal structure of many important C*-algebras, such as transformation group algebras with abelian stabilizers and higher-rank graph algebras, is accessible through explicit topological models and combinatorial data.

Abstract

Given a Hausdorff locally compact étale groupoid $\mathcal G$, we describe as a topological space the part of the primitive spectrum of $C^*(\mathcal G)$ obtained by inducing one-dimensional representations of amenable isotropy groups of $\mathcal G$. When $\mathcal G$ is amenable, second countable, with abelian isotropy groups, our result gives the description of $\operatorname{Prim} C^*(\mathcal G)$ conjectured by van Wyk and Williams. This, in principle, completely determines the ideal structure of a large class of separable C$^*$-algebras, including the transformation group C$^*$-algebras defined by amenable actions of discrete groups with abelian stabilizers and the C$^*$-algebras of higher rank graphs. As an illustration we describe the primitive spectrum of the C$^*$-algebra of any row-finite higher rank graph without sources.

The primitive spectrum of C*-algebras of etale groupoids with abelian isotropy

TL;DR

This work provides a topological description of the primitive spectrum for amenable, second countable, étale groupoids with abelian isotropy by identifying it with a quotient of the character space via induction from isotropy. The authors introduce the spaces and , establish a robust induction map, and prove that, under the stated hypotheses, yields a homeomorphism from onto . They develop (i) a general framework for injectively graded groupoids, (ii) concrete results for transformation groupoids, Deaconu–Renault groupoids, and (iii) explicit descriptions for graph algebras, including 1-graphs and higher-rank graphs. Consequently, the ideal structure of many important C*-algebras, such as transformation group algebras with abelian stabilizers and higher-rank graph algebras, is accessible through explicit topological models and combinatorial data.

Abstract

Given a Hausdorff locally compact étale groupoid , we describe as a topological space the part of the primitive spectrum of obtained by inducing one-dimensional representations of amenable isotropy groups of . When is amenable, second countable, with abelian isotropy groups, our result gives the description of conjectured by van Wyk and Williams. This, in principle, completely determines the ideal structure of a large class of separable C-algebras, including the transformation group C-algebras defined by amenable actions of discrete groups with abelian stabilizers and the C-algebras of higher rank graphs. As an illustration we describe the primitive spectrum of the C-algebra of any row-finite higher rank graph without sources.
Paper Structure (17 sections, 43 theorems, 86 equations)

This paper contains 17 sections, 43 theorems, 86 equations.

Table of Contents

  1. Preliminaries
  2. Étale groupoids and their C*-algebras
  3. Weak containment and the Jacobson topology
  4. Fell topology
  5. T0-ization
  6. Primitive ideals induced by characters
  7. The spaces Char(G), Stab(G) and the induction map
  8. Main results
  9. Injectively graded groupoids
  10. Transformation groupoids
  11. Groupoids graded by abelian groups
  12. Harmonious families of bisections
  13. Deaconu--Renault groupoids
  14. Singly generated dynamical systems
  15. Graph algebras
  16. ...and 2 more sections

Key Result

Lemma 1.1

For every $g\in\mathcal{G}_x$, the right translation by $g$ defines a unitary equivalence between the representations $\operatorname{Ind}^\mathcal{G}_S\pi$ and $\operatorname{Ind}^\mathcal{G}_{gSg^{-1}}\pi(g^{-1}\cdot g)$.

Theorems & Definitions (94)

  • Lemma 1.1
  • proof
  • Lemma 1.2: MR0146681*Theorem 1.4
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • proof
  • Corollary 1.5
  • proof
  • Corollary 1.6
  • ...and 84 more