Quotient ideal amenability of Banach algebras

Abstract

‎The concepts of amenability and weak amenability of Banach algebras have a relatively long history‎, ‎which were introduced In 1972 B‎. ‎E‎. ‎Johnson in \cite{BEJ}‎, ‎and the class of weakly amenable Banach algebra is considerably larger than that of amenable Banach algebras‎. ‎Let $A$ be a Banach algebra and $X$ be a Banach $A$-bimodule‎, ‎$X^*$ becomes a Banach $A$-bimodule with following actions‎ ‎\[‎ ‎\langle x,a\cdot f\rangle=\langle x\cdot a,f\rangle \quad,\quad \langle x,f\cdot a\rangle=\langle a\cdot x,f\rangle \qquad (a\in A‎ , ‎f\in X^*‎ , ‎x\in X)‎. ‎\]‎ ‎The continuous map $D:A\to X$ is called a derivation if $D(a_1 a_2)=a_1\cdot D(a_2)+D(a_1)\cdot a_2$ for all $a_1,a_2\in A$ and is called inner derivation if exist $x\in X$ such that $D(a)=a\cdot x‎- ‎x\cdot a\ \ (a\in A)$ and in this case‎, ‎we denote the inner derivation $D$ dependent on the element $x$ with $\textbf{ad}_x$‎. ‎The derivation $D:A\to X$ is bounded if there exists a constant $M>0$ such that‎, ‎$\parallel D(a) \parallel \leq M \parallel a \parallel$ for each $a \in A$‎. ‎We denote the linear space of all derivations from $A$ into $X$ by $\mathcal{Z}^1(A‎, ‎X)$ and the linear space of inner derivations from $A$ into $X$ by $\mathcal{B}^1(A‎, ‎X)$‎. ‎The first cohomology group with coefficients in $X$ is denoted by $\mathcal{H}^1(A‎, ‎X)$ which is the quotient group $\mathcal{Z}^1(A‎, ‎X)/\mathcal{B}^1(A‎, ‎X)$‎.