Scattered locally C*-algebras

TL;DR

This paper introduces the concept of scattered locally C*-algebras, provides conditions for their characterization, and explores when crossed products with group actions preserve the scattered property.

Contribution

It defines scattered locally C*-algebras and investigates conditions under which their crossed products remain scattered.

Findings

Conditions for a locally C*-algebra to be scattered

Results on when crossed products of scattered algebras are scattered

Insights into group actions on scattered locally C*-algebras

Abstract

In this paper, we introduce the notion of a scattered locally $C^{\ast }$ -algebra and we give conditions for a locally $C^{\ast }$-algebra to be scattered. Given an action $\alpha $ of a locally compact group $G$ on a scattered locally $C^{\ast }$-algebra $A[\tau_{\Gamma }]$, it is natural to ask under what conditions the crossed product $A[\tau_{\Gamma }]\times_{\alpha }G$ is also scattered. We obtain some results concerning this question.