  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.
