### 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 $A_i$

($i=1,2$).

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 $A_i$

($i=1,2$).

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