Scattered locally C*-algebras

Abstract

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