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A going-down principle for {é}tale groupoids and the Baum-Connes conjecture

Kai Mao

Abstract

We study a going-down principle for {é}tale groupoids and its applications, extending the earlier results for locally compact groups by Chabert, Echterhoff and Oyono-Oyono, and for ample groupoids by B{ö}nicke and by B{ö}nicke-Dell'Aiera. The proof in the general {é}tale groupoid setting is based on a more detailed study of groupoid simplicial complexes. We also study a bicategorical functoriality involving the induction functors from {é}tale groupoid correspondences, which was introduced by Miller. This yields a bicategorical interpretation of the induction-restriction adjunction. As an application of the going-down principle, we provide a proof of the split injectivity of Baum-Connes assembly map for {é}tale groupoids that are strongly amenable at infinity, recovering a result obtained by B{ö}nicke and Proietti via a categorical approach. The going-down principle is also applied on the proof of continuity of topological K-theory of {é}tale groupoids and the study of scope of validity of K{ü}nneth formulas.

A going-down principle for {é}tale groupoids and the Baum-Connes conjecture

Abstract

We study a going-down principle for {é}tale groupoids and its applications, extending the earlier results for locally compact groups by Chabert, Echterhoff and Oyono-Oyono, and for ample groupoids by B{ö}nicke and by B{ö}nicke-Dell'Aiera. The proof in the general {é}tale groupoid setting is based on a more detailed study of groupoid simplicial complexes. We also study a bicategorical functoriality involving the induction functors from {é}tale groupoid correspondences, which was introduced by Miller. This yields a bicategorical interpretation of the induction-restriction adjunction. As an application of the going-down principle, we provide a proof of the split injectivity of Baum-Connes assembly map for {é}tale groupoids that are strongly amenable at infinity, recovering a result obtained by B{ö}nicke and Proietti via a categorical approach. The going-down principle is also applied on the proof of continuity of topological K-theory of {é}tale groupoids and the study of scope of validity of K{ü}nneth formulas.
Paper Structure (34 sections, 116 theorems, 307 equations)

This paper contains 34 sections, 116 theorems, 307 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Groupoids and actions
  4. Local structure of proper actions of étale groupoids
  5. C0(X)-algebras and modules
  6. Groupoid C*-algebras and Hilbert modules
  7. Étale groupoid correspondences
  8. Induction
  9. Homomorphism tau
  10. Groupoid pre-Hilbert modules
  11. Groupoid simplicial complexes
  12. Space of measures
  13. Groupoid simplicial complexes
  14. Topological properties of groupoid simplicial complexes
  15. Rips complexes
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $\mathcal{G}$ be a second countable locally compact Hausdorff étale groupoid, $A,B$ be separable $\mathcal{G}$-$C^*$-algebra. If $x\in \mathrm{KK}^\mathcal{G}_0(A,B)$ such that is an isomorphism for any proper open subgroupoid $\mathcal{H}\subseteq \mathcal{G}$. Then is an isomorphism.

Theorems & Definitions (237)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Example 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 227 more