A metric approach to limit operators

TL;DR

This paper generalizes the limit operator theory from lattice groups to discrete metric spaces with bounded geometry, establishing a criterion for Fredholmness of band-dominated operators based on invertibility of limit operators, contingent on property A.

Contribution

It extends the limit operator framework to metric spaces without group actions, linking Fredholmness to property A and invertibility of limit operators.

Findings

Fredholmness characterized by invertibility of all limit operators for spaces with property A.

Failure of the characterization for spaces lacking property A.

Generalization of limit operator machinery to non-group metric spaces.

Abstract

We extend the limit operator machinery of Rabinovich, Roch, and Silbermann from $\mathbb{Z}^N$ to (bounded geometry, strongly) discrete metric spaces. We do not assume the presence of any group structure or action on our metric spaces. Using this machinery and recent ideas of Lindner and Seidel, we show that if a metric space X has Yu's property A, then a band-dominated operator on X is Fredholm if and only if all of its limit operators are invertible. We also show that this always fails for metric spaces without property A.