Uniform Local Amenability

TL;DR

This paper introduces uniform local amenability, a new property linking coarse geometric properties, and demonstrates its implications for operator norm estimates and the relationship between amenability and asymptotic dimension.

Contribution

It defines the novel property of uniform local amenability and explores its implications for coarse geometry and operator theory, including generalizing a theorem of Nowak.

Findings

Property A implies operator norm localisation.

Uniform local amenability helps identify 'bad' spaces.

Generalization of Nowak's theorem on amenability and asymptotic dimension.

Abstract

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. We also generalise and reprove a theorem of Nowak relating amenability and asymptotic dimension in the quantitative setting.