  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.
