Uniform local amenability

Abstract

The main results of this paper show that various coarse ('large scale') geometric properties are closely related. In particular, we show that propertyA implies the operator norm localisation property, and thus that norms of operators associated to a very large class of metric spaces can be effectively estimated. The main tool is a new property called uniform local amenability. This property is easy to negate, which we use to study some 'bad' spaces: specifically, expanders and graphs with large girth. We also generalise and reprove a theorem of Nowak relating amenability and asymptotic dimension in the quantitative setting. © European Mathematical Society.