Uniform local amenability

The main results of this paper show that various coarse ('large scale') geometric properties are closely related. In particular, we show that property A 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.