# Topological centers of module actions and cohomological groups of Banach
Algebras

Research paper by **Kazem Azem Haghnejad Azar**

Indexed on: **16 Aug '10**Published on: **16 Aug '10**Published in: **Mathematics - Functional Analysis**

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#### 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 $L_f(g)=f.g$.