Amenable groups and bounded $\Delta$-weak approximate identities

Abstract

Let $A$ be a Banach algebra with a non-empty character space. We say that a bounded net $\{e_{\alpha }\}$ in $A$ is a bounded $\Delta$-weak approximate identity for $A$ if, for each $a\in A$ and compact subset $K$ of $\Delta (A)$, $||\widehat{e_{\alpha }a}-\hat{a}|{|}_K={\sup }_{\varphi \in K}|\varphi (e_{\alpha }a)-\varphi (a)|\to 0$. For each $1<p<\infty$, we prove that the Figa-Talamanca Herz algebra, $A_p(G)$ has a bounded $\Delta$-weak approximate identity if and only if $G$ is an amenable group. Also we give a sufficient condition for amenability of group $G$.

arXiv comment6 pages