Topological and frame properties of certain pathological $C^*$-algebras

TL;DR

This paper classifies certain topological spaces based on their $\sigma$-compact subsets and explores how these classifications influence the properties of associated $C^*$-algebras, including frame theory and $\mathcal A$-compact operators, with some pathological examples.

Contribution

It introduces a new classification of topological spaces and analyzes the resulting $C^*$-algebra properties using frame theory and $\mathcal A$-compact operators, including constructing pathological examples.

Findings

Classification of spaces based on $\sigma$-compact subsets.

Analysis of $C^*$-algebra properties via frame theory.

Construction of pathological $C^*$-algebra examples.

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 $C^*$-algebras in terms of frame theory and the theory of $\mathcal A$-compact operators in Hilbert $C^*$-modules, some pathological examples are constructed.