On Weak generalized amenability of triangular Banach algebras

Abstract

Let $A_1$, $A_2$ be unital Banach algebras and $X$ be an $A_1-A_2-$
module. Applying the concept of module maps, (inner) module
generalized derivations and generalized first cohomology groups, we
present several results concerning the relations between module
generalized derivations from $A_i$ into the dual space $A^*_i$ (for
$i=1,2$) and such derivations from the triangular Banach algebra
of the form $\mathcal{T} :=\left(\begin{array}{lc}
A_1 &X\\
0 & A_2\end{array}\right)$ into the associated triangular $\mathcal{T}-$ bimodule $\mathcal{T}^*$ of the
form $\mathcal{T}^*:=\left(\begin{array}{lc}
A_1^* &X^*\\
0 & A_2^*\end{array}\right)$. In particular, we show that the so-called generalized first
cohomology group from $\mathcal{T}$ to $\mathcal{T}^*$ is isomorphic to the directed sum of the generalized first
cohomology group from $A_1$ to $A^*_1$ and the generalized first
cohomology group from $A_2$ to $A_2^*$. Finally, Inspiring the above concepts, we
establish a one to one corresponding between weak (resp. ideal) generalized
amenability of $\mathcal{T}$ and those amenability of $A_i$
($i=1,2$).