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Equivariant periodic cyclic homology for ample groupoids

Francesco Pagliuca, Christian Voigt

Abstract

We define and study bivariant equivariant periodic cyclic homology for actions of ample groupoids. In analogy to the group case, we show that the theory satisfies homotopy invariance, stability, and excision in both variables. We also prove an analogue of the Green-Julg theorem for actions of proper groupoids.

Equivariant periodic cyclic homology for ample groupoids

Abstract

We define and study bivariant equivariant periodic cyclic homology for actions of ample groupoids. In analogy to the group case, we show that the theory satisfies homotopy invariance, stability, and excision in both variables. We also prove an analogue of the Green-Julg theorem for actions of proper groupoids.

Paper Structure

This paper contains 28 sections, 47 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Groupoids
  4. Algebras and multipliers
  5. Function spaces
  6. Proper groupoids
  7. The category of $\mathcal{G}$-modules
  8. $\mathcal{G}$-modules
  9. Tensor products
  10. $\mathcal{G}$-algebras
  11. Algebras of functions
  12. Algebras associated with pairings
  13. $C_c^\infty(X)$-algebras
  14. Unitarisation
  15. Crossed products
  16. ...and 13 more sections

Key Result

Lemma 2.2

Let $f: A \rightarrow M(B)$ be an essential algebra homomorphism. If the multiplication in $B$ is nondegenerate there exists a unique unital algebra homo-morphism $F: M(A) \rightarrow M(B)$ such that $F \iota = f$.

Theorems & Definitions (98)

  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 88 more