A nonstanadard analysis approach to limit operators and Fredholmness in Roe-like algebras

TL;DR

This paper introduces a nonstandard analysis approach to characterize Fredholm operators in Roe-like algebras via limit operators on galaxies, improving existing criteria by reducing the number of limit operators needed.

Contribution

It develops a novel nonstandard analysis framework for limit operators in Roe-like algebras, providing a more efficient characterization of Fredholmness.

Findings

Fredholmness characterized by invertibility of limit operators on galaxies.

Invariance of Fredholm property under nonstandard limit operators.

Strengthens previous results by reducing the necessary limit operators.

Abstract

Let $(X,d)$ be a uniformly locally finite metric space, and $T$ an operator in the uniform Roe algebra $C_u^*(X)$ (or uniform quasi-local algebra $C_{ql}^*(X)$). In this paper, we introduce the concept of limit operators of $T$ on galaxies in the nonstandard extension of $X$, and prove that $T$ is a generalized Fredholm operator with respect to the ghost ideal in $C_u^*(X)$ (or $C_{ql}^*(X)$) if and only if all limit operators on afar galaxies are invertible, and their inverses are uniformly bounded. In particular, if $X$ has Yu's Property A, then $T$ is a Fredholm operator if and only if all limit operators on afar galaxies are invertible. Using techniques in nonstandard analysis, our result strengthens a work of \v{S}pakula--Willett \cite{SpW} on the characterization of Fredholmness by using less limit operators.