Harmonic functionals on certain Banach algebras

Abstract

In this paper, we study the concept of harmonic functionals for certain Banach algebras such as generalized Fourier algebras. For a non-zero character $\phi$ on Banach algebra ${\mathcal A}$, we also characterize the concept of $\phi$-amenability in terms of harmonic functionals. Finally, for a locally compact group $G$ we investigate the space $H_{\sigma, x}$ of $\sigma$-harmonic functionals in the dual of generalized Fourier algebra $A_p(G)$. The main result states that $G$ is first countable if and only if $\sigma$ is adapted if and only if $H_{\sigma, x}={\Bbb C}\phi_x$.