The Topological Centers Of Module Actions

In this article, for Banach left and right module actions, we will extend some propositions from Lau and $\ddot{U}lger$ into general situations and we establish the relationships between topological centers of module actions. We also introduce the new concepts as $Lw^*w$-property and $Rw^*w$-property for Banach $A-bimodule$ $B$ and we investigate the relations between them and topological center of module actions. We have some applications in dual groups.Comment: 15 pag

The topological centers and factorization properties of module actions and ∗−involution\ast-involution algebras

For Banach left and right module actions, we extend some propositions from Lau and $\ddot{U}lger$ into general situations and we establish the relationships between topological centers of module actions. We also introduce the new concepts as $Lw^*w$-property and $Rw^*w$-property for Banach $A-bimodule$ $B$ and we obtain some conclusions in the topological center of module actions and Arens regularity of Banach algebras. we also study some factorization properties of left module actions and we find some relations of them and topological centers of module actions. For Banach algebra $A$, we extend the definition of $\ast-involution$ algebra into Banach $A-bimodule$ $B$ with some results in the factorizations of $B^*$. We have some applications in group algebras

Derivations And Cohomological Groups Of Banach Algebras

Let $B$ be a Banach $A-bimodule$ and let $n\geq 0$. We investigate the relationships between some cohomological groups of $A$, that is, if the topological center of the left module action $\pi_\ell:A\times B\rightarrow B$ 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 $H^1(A,B^*)$ 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 $H^1_{w^*}(L^\infty(G)^*,L^\infty(G)^{**})=0$. Let $G$ be an amenable locally compact group. Then there is a Banach $L^1(G)-bimodule$ such as $(L^\infty(G),.)$ such that $Z^1(L^1(G),L^\infty(G))=\{L_{f}:~f\in L^\infty(G)\}.$ We also obtain some conclusions in the Arens regularity of module actions and weak amenability of Banach algebras. We introduce some new concepts as $left-weak^*-to-weak$ convergence property [$=Lw^*wc-$property] and $right-weak^*-to-weak$ convergence property [$=Rw^*wc-$property] with respect to $A$ and we show that if $A^*$ and $A^{**}$, respectively, have $Rw^*wc-$property and $Lw^*wc-$property and $A^{**}$ is weakly amenable, then $A$ is weakly amenable. We also show to relations between a derivation $D:A\rightarrow A^*$ and this new concepts

Arens regularity and weak topological center of module actions

Let $A$ be a Banach algebra and $A^{**}$ be the second dual of it. We define $\tilde{Z}_1(A^{**})$ as a weak topological center of $A^{**}$ with respect to first Arens product and we will find some relations between this concept and the topological center of $A^{**}$. We also extend this new definition into the module actions and find relationship between weak topological center of module actions and reflexivity or Arens regularity of some Banach algebras, and we investigate some applications of this new definition in the weak amenability of some Banach algebras