Left introverted subspaces of duals of Banach algebras and $WEA{K}^{\ast }-$continuous derivations on dual Banach algebras

Abstract

Let $X$ be a left introverted subspace of dual of a Banach algebra. We study $Z_t(X^{\ast }),$ the topological center of Banach algebra $X^{\ast }$. We fined the topological center of $(X\cA)^*$, when $\cA$ has a bounded right approximate identity and $\cA\subseteq X^*.$ So we introduce a new notation of amenability for a dual Banach algebra $\mathcal{A}$. A dual Banach algebra $\mathcal{A}$ is weakly Connes-amenable if the first $wea{k}^{\ast }-$continuous cohomology group of $\mathcal{A}$ with coefficients in $\mathcal{A}$ is zero; i.e., $H_{w^{\ast }}^1(\mathcal{A},\mathcal{A})=\{o\}$. We study the weak Connes-amenability of some dual Banach algebras.

arXiv comment10 pages