On $\Delta$-weak $\phi$-amenability of Banach algebras

Abstract

Let $A$ be a Banach algebra and $\phi\in \Delta(A)\cup\{0\}$. We say that $A$ is $\Delta$-weak $\phi$-amenable if there exists an $m\in A^{**}$ such that $m(\phi)=0$ and $m(\psi.a)=\psi(a)$ for each $\psi\in \Delta(A)$ and $a\in \ker(\phi)$. It is shown that $A$ is $\Delta$-weak $\phi$-amenable if and only if $\ker(\phi)$ has a bounded $\Delta$-weak approximate identity. We examine this notion for some algebras over amenable locally compact groups. Also we prove that every $\Delta$-weak $\phi$-amenable Banach algebra has a bounded $\Delta$-weak approximate identity.