For a bounded linear operator on $\ell^2(\mathbb{Z})$ it is well-known and relatively easy to prove that it is compact if and only if it is band-dominated and all limit operators are $0$. In 2017, \v{S}pakula and Willett [1] showed that this generalizes to $\ell^2(X)$, where $X$ is a strongly discrete metric space of bounded geometry that satisfies Yu's property A. Maybe more interestingly, \v{S}pakula and Willett also showed that property A is actually necessary in the sense that if $X$ is a strongly discrete metric space of bounded geometry that does not satisfy property A, then there exist non-compact operators for which all limit operators are $0$. This particularly revealed some intriguing connections between operator theory and coarse geometry. While it is not so easy to construct a space that does not satisfy property A, it is quite easy to see that the above compactness characterization fails for non-discrete spaces. Even without stating precise definitions, in every reasonable sense a function of compact support will vanish if shifted to infinity and hence all limit operators of the corresponding multiplication operator also vanish. On the other hand, a multiplication operator on $L^2(\mathbb{R})$ is almost never compact. In this talk we will review the discrete case and then discuss the proper generalization to non-discrete lca groups, which appears quite natural from a QHA point of view. Moreover, I will explain the geometric obstruction called property A and why it doesn't occur in this setting.
Based on joint work with Robert Fulsche.
[1] J. \v{S}pakula and R. Willett: \emph{A metric approach to limit operators}, Trans. Am. Math. Soc. 369, 263–308, 2017.