Operator norm localization property for equi-approximable families of projections

TL;DR

This paper introduces weaker operator norm localization properties that characterize geometric conditions in uniform Roe algebras, leading to new rigidity results and insights into their structure and embeddings.

Contribution

It defines weaker ONL properties that characterize geometric conditions, advancing understanding of rigidity and structure in uniform Roe algebras.

Findings

Weaker ONL properties characterize certain geometric conditions.

New rigidity results for nonmetrizable coarse spaces.

Partial answer to White and Willett's question on Cartan subalgebras.

Abstract

The rigidity problem for uniform Roe algebras was recently positively solved. Before its solution was found, there were positive solutions under the assumption of certain technical geometric conditions. In this paper, we introduce weaker versions of the operator norm localization property (ONL) which turn out to characterize those technical geometric conditions. We use this to obtain new rigidity results for nonmetrizable coarse spaces. As an application, we provide a novel partial answer to a question of White and Willett about Cartan subalgebras of uniform Roe algebras. We also study embeddings between uniform Roe algebras.