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

Let $(X,d)$ be a uniformly locally finite metric space, and $T$ an operator in the uniform Roe algebra $C_u^{\ast }(X)$ (or uniform quasi-local algebra $C_{ql}^{\ast }(X)$). 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 $C_u^{\ast }(X)$ (or $C_{ql}^{\ast }(X)$) 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 Špakula–Willett \cite{SpW} on the characterization of Fredholmness by using less limit operators.