A note on inner amenability for FLC point sets

Abstract

Inner amenability is a bridge between amenability of an object and amenability of its operator algebras. It is an open problem of Ananantharman-Delaroche to decide whether all \'etale groupoids are inner amenable. Approximate lattices and their dynamics have recently attracted increased attention and have been studied using groupoid methods. In this note, we prove that groupoids associated with approximate lattices in second countable locally compact groups are inner amenable. In fact we show that this result holds more generally for point sets of finite local complexity in such groups.