Topological centers of module actions and cohomological groups of Banach Algebras

TL;DR

This paper investigates the topological centers of module actions and cohomological groups of Banach algebras, establishing relationships between Arens regularity, module actions, and properties like Connes-amenability, with applications to group algebras.

Contribution

It extends existing results on Arens regularity and cohomology of Banach algebras, providing new insights into their topological centers and cohomological properties, especially for group algebras.

Findings

Topological centers of module actions relate to Arens regularity.

For certain Banach modules, centers coincide with original spaces.

Established vanishing of specific cohomology groups for compact groups.

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 $L_f(g)=f.g$.