Topological centers of module actions and cohomological groups of Banach Algebras (1008.2655v1)

Abstract: In this paper, first we study some Arens regularity properties of module actions. Let $B$ be a Banach $A-bimodule$ and let ${Z}^\ell_{B^{}}(A^{})$ and ${Z}^\ell_{A^{}}(B^{})$ be the topological centers of the left module action $\pi_\ell:~A\times B\rightarrow B$ and the right module action $\pi_r:~B\times A\rightarrow B$, respectively. We investigate some relationships between topological center of $A^{**}$, ${Z}1({A^{**}})$ with respect to the first Arens product and topological centers of module actions ${Z}^\ell{B^{}}(A^{})$ and ${Z}^\ell_{A^{}}(B^{})$. On the other hand, if $A$ has Mazure property and $B^{**}$ has the left $A^{**}-factorization$, then $Z^\ell_{A^{}}(B^{})=B$, and so for a locally compact non-compact group $G$ with compact covering number $card(G)$, we have $Z^\ell_{M(G)^{}}{(L^1(G)^{})}= {L^1(G)}$ and $Z^\ell_{L^1(G)^{}}{(M(G)^{})}= {M(G)}$. 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 $H^1_{w^}(L^\infty(G)^,L^\infty(G)^{**})=0$. Suppose that $G$ is an amenable locally compact group. Then there is a Banach $L^1(G)-bimodule$ such as $(L^\infty(G),.)$ such that $Z^1(L^1(G),L^\infty(G))={L_{f}:~f\in L^\infty(G)}$ where for every $g\in L^1(G)$, we have $L_f(g)=f.g$.