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

Abstract

Let $A$ be a Banach algebra and $\varphi \in \Delta (A)\cup \{0\}$. We say that $A$ is $\Delta$-weak $\varphi$-amenable if there exists an $m\in A^{\ast \ast }$ such that $m(\varphi )=0$ and $m(\psi .a)=\psi (a)$ for each $\psi \in \Delta (A)$ and $a\in \ker (\varphi )$. It is shown that $A$ is $\Delta$-weak $\varphi$-amenable if and only if $\ker (\varphi )$ 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 $\varphi$-amenable Banach algebra has a bounded $\Delta$-weak approximate identity.

11 pages, some typos corrected, the converse of a theorem added and a minor revision in abstract