Quantum metrics from length functions on \'etale groupoids

TL;DR

This paper develops a method to construct compact quantum metric spaces from length functions on étale groupoids, with applications to AF groupoids and unital AF algebras.

Contribution

It introduces a new approach to quantum metric geometry using groupoid length functions, including necessary and sufficient conditions and applications to AF algebras.

Findings

Constructed compact quantum metric spaces from étale groupoids with length functions.

Provided a sufficient condition for the resulting space to be a compact quantum metric space.

Showed that any AF groupoid with a compact unit space admits such a length function.

Abstract

We show how to construct a compact quantum metric space from a proper continuous length function on an \'etale 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 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.