Invariant $\varphi$-means for abstract Segal algebras related to locally compact groups

Abstract

In this paper, for a locally compact group ${\mathcal G}$ we characterize character amenability and character contractibility of abstract Segal algebras with respect to the group algebra $L^1({\mathcal G})$ and the generalized Fourier algebra $A_p({\mathcal G})$. As a main result we prove that ${\mathcal G}$ is discrete and amenable if and only if some class of abstract Segal algebras in $L^1({\mathcal G})$ is character amenable. We also prove a similar result for abstract Segal algebras in $A_p({\mathcal G})$ and $C_0({\mathcal G})$. Finally, under some conditions we investigate when a commutative, semisimple, Tauberian Banach algebra is an ideal in its second dual space.