A nonstandard analysis approach to limit operators and Fredholmness in Roe‐like algebras

Abstract

Let be a uniformly locally finite metric space, and an operator in the uniform Roe algebra (or uniform quasi‐local algebra ). In this paper, we introduce the concept of limit operators of on galaxies in the nonstandard extension of , and prove that is a generalized Fredholm operator with respect to the ghost ideal in (or ) if and only if all limit operators on afar galaxies are invertible, and their inverses are uniformly bounded. In particular, if has Yu's Property A, then 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 [Trans. Amer. Math. Soc. 369 (2017), no. 1, 263–308] on the characterization of Fredholmness by using fewer limit operators.