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Polynomial growth and property $RD_p$ for étale groupoids with applications to $K$-theory

Are Austad, Eduard Ortega, Mathias Palmstrøm

Abstract

We investigate property $RD_p$ for étale groupoids and apply it to $K$-theory of reduced groupoid $L^p$-operator algebras. In particular, under the assumption of polynomial growth, we show that the $K$-theory groups for a reduced groupoid $L^p$-operator algebra are independent of $p\in (1, \infty)$. We apply the results to coarse groupoids and graph groupoids.

Polynomial growth and property $RD_p$ for étale groupoids with applications to $K$-theory

Abstract

We investigate property for étale groupoids and apply it to -theory of reduced groupoid -operator algebras. In particular, under the assumption of polynomial growth, we show that the -theory groups for a reduced groupoid -operator algebra are independent of . We apply the results to coarse groupoids and graph groupoids.
Paper Structure (11 sections, 37 theorems, 148 equations)

This paper contains 11 sections, 37 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $K$-theory for Fréchet Algebras
  4. Étale Groupoids
  5. Reduced Groupoid $L^p$-Operator Algebras
  6. Rapid Decay and Polynomial Growth
  7. Applications
  8. Examples
  9. Coarse Groupoids
  10. Point Set Groupoids
  11. Graph Groupoids

Key Result

Theorem A

Let $\mathcal{G}$ be an étale groupoid endowed with a continuous length function for which it has property $RD_p$ and $RD_q$, where $p,q \in (1, \infty)$ are Hölder conjugate. Then

Theorems & Definitions (73)

  • Theorem A: cf. \ref{['thm: RD_q and RD_p implies isomorphisms in K-theory']}
  • Theorem B: cf. \ref{['thm: Polynomial growth implies all K-groups are isomorphic']}
  • Theorem C: cf. \ref{['corollary:roe-algebra-k-theory-results']}
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4: CGTRigidityResultsForLpOp
  • Proposition 2.5: CGTRigidityResultsForLpOp
  • Lemma 2.6
  • ...and 63 more