Derivations And Cohomological Groups Of Banach Algebras

Let $B$ be a Banach $A-bimodule$ and let $n\ge 0$. We investigate the relationships between some cohomological groups of $A$, that is, if the topological center of the left module action ${\pi }_{\mathrm{\ell }}:A\times B\to B$ of $A^{(2n)}$ on $B^{(2n)}$ is $B^{(2n)}$ and $H^1(A^{(2n+2)},B^{(2n+2)})=0$, then we have $H^1(A,B^{(2n)})=0$, and we find the relationships between cohomological groups such as $H^1(A,B^{(n+2)})$ and $H^1(A,B^{(n)})$, spacial $H^1(A,B^{\ast })$ and $H^1(A,B^{(2n+1)})$. We obtain some results in Connes-amenability of Banach algebras, and so for every compact group $G$, we conclude that $H_{w^{\ast }}^1(L^{\mathrm{\infty }}(G{)}^{\ast },L^{\mathrm{\infty }}(G{)}^{\ast \ast })=0$. Let $G$ be an amenable locally compact group. Then there is a Banach $L^1(G)-bimodule$ such as $(L^{\mathrm{\infty }}(G),\mathrm{.})$ such that $Z^1(L^1(G),L^\infty(G))={L_{f}:~f\in L^\infty(G)}.$ We also obtain some conclusions in the Arens regularity of module actions and weak amenability of Banach algebras. We introduce some new concepts as $left-wea{k}^{\ast }-to-weak$ convergence property [$=L{w}^{\ast }wc-$property] and $right-wea{k}^{\ast }-to-weak$ convergence property [$=R{w}^{\ast }wc-$property] with respect to $A$ and we show that if $A^{\ast }$ and $A^{\ast \ast }$, respectively, have $R{w}^{\ast }wc-$property and $L{w}^{\ast }wc-$property and $A^{\ast \ast }$ is weakly amenable, then $A$ is weakly amenable. We also show to relations between a derivation $D:A\to A^{\ast }$ and this new concepts.