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Quantum metrics from length functions on étale groupoids

Are Austad

Abstract

We show how to construct a compact quantum metric space from a proper continuous length function on an étale groupoid with compact unit space, where the unit space additionally has the structure of a compact metric space. Using compactly supported Fourier multipliers on the reduced groupoid $C^*$-algebra we provide a sufficient condition for verifying when we obtain a compact quantum metric space in this manner. The condition is sometimes also necessary, and is new even in the case of length functions on discrete groups. Lastly, we employ the condition to show that any AF groupoid with compact unit space can be equipped with a length function from which we obtain a compact quantum metric space, thereby providing a groupoid approach to understanding the quantum metric geometry of unital AF algebras.

Quantum metrics from length functions on étale groupoids

Abstract

We show how to construct a compact quantum metric space from a proper continuous length function on an étale groupoid with compact unit space, where the unit space additionally has the structure of a compact metric space. Using compactly supported Fourier multipliers on the reduced groupoid -algebra we provide a sufficient condition for verifying when we obtain a compact quantum metric space in this manner. The condition is sometimes also necessary, and is new even in the case of length functions on discrete groups. Lastly, we employ the condition to show that any AF groupoid with compact unit space can be equipped with a length function from which we obtain a compact quantum metric space, thereby providing a groupoid approach to understanding the quantum metric geometry of unital AF algebras.
Paper Structure (9 sections, 21 theorems, 143 equations)

This paper contains 9 sections, 21 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Groupoids
  4. Multipliers of reduced groupoid $C^*$-algebras
  5. Compact quantum metric spaces
  6. Compact quantum metric spaces from groupoids
  7. Designing the seminorm and operator system
  8. A characterization of metric groupoids yielding compact quantum metric spaces
  9. AF groupoids

Key Result

Proposition 2.4

Any $C \in \mathrm{FS}(\mathcal{G})$ gives rise to a completely bounded map $m_C \colon C_{r}^{*} (\mathcal{G}) \to C_{r}^{*}(\mathcal{G})$ given by $m_C(f) = C \cdot f$, where $(C \cdot f)(\gamma) = C(\gamma)f(\gamma)$ for $f \in C_{r}^{*} (\mathcal{G})$. Moreover, the completely bounded multiplier

Theorems & Definitions (61)

  • Remark 2.1
  • Definition 2.2
  • Proposition 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • Proposition 2.8
  • Remark 2.9
  • Definition 2.10
  • ...and 51 more