Topological and frame properties of certain pathological $C^{\ast }$-algebras

Abstract

We introduce a classification of locally compact Hausdorff topological spaces with respect to the behavior of $\sigma$-compact subsets, and relying on this classification we study properties of corresponding $C^{\ast }$-algebras in terms of frame theory and the theory of $\mathcal{A}$-compact operators in Hilbert $C^{\ast }$-modules, some pathological examples are constructed.