Topological centers of module actions and cohomological groups of Banach Algebras

In this paper, first we study some Arens regularity properties of module actions. Let $B$ be a Banach $A-bimodule$ and let $Z_{B^{\ast \ast }}^{\mathrm{\ell }}(A^{\ast \ast })$ and $Z_{A^{\ast \ast }}^{\mathrm{\ell }}(B^{\ast \ast })$ be the topological centers of the left module action ${\pi }_{\mathrm{\ell }}:A\times B\to B$ and the right module action ${\pi }_r:B\times A\to B$, respectively. We investigate some relationships between topological center of $A^{\ast \ast }$, $Z_1(A^{\ast \ast })$ with respect to the first Arens product and topological centers of module actions $Z_{B^{\ast \ast }}^{\mathrm{\ell }}(A^{\ast \ast })$ and $Z_{A^{\ast \ast }}^{\mathrm{\ell }}(B^{\ast \ast })$. On the other hand, if $A$ has Mazure property and $B^{\ast \ast }$ has the left $A^{\ast \ast }-factorization$, then $Z_{A^{\ast \ast }}^{\mathrm{\ell }}(B^{\ast \ast })=B$, and so for a locally compact non-compact group $G$ with compact covering number $card(G)$, we have $Z_{M(G{)}^{\ast \ast }}^{\mathrm{\ell }}(L^1(G{)}^{\ast \ast })=L^1(G)$ and $Z_{L^1(G{)}^{\ast \ast }}^{\mathrm{\ell }}(M(G{)}^{\ast \ast })=M(G)$. By using the Arens regularity of module actions, we study some cohomological groups properties of Banach algebra and we extend some propositions from Dales, Ghahramani, Gr{\o}nb{\ae}k and others into general situations and we investigate the relationships between some cohomological groups of Banach algebra $A$. We obtain some results in Connes-amenability of Banach algebras, and so for every compact group $G$, we conclude that $H_{w^{\ast }}^1(L^{\mathrm{\infty }}(G{)}^{\ast },L^{\mathrm{\infty }}(G{)}^{\ast \ast })=0$. Suppose that $G$ is an amenable locally compact group. Then there is a Banach $L^1(G)-bimodule$ such as $(L^{\mathrm{\infty }}(G),\mathrm{.})$ such that $Z^1(L^1(G),L^\infty(G))={L_{f}:~f\in L^\infty(G)}$whereforevery$g\in L^1(G)$,wehave$L_f(g)=f.g$.