Lie Weak Amenability of Triangular Banach Algebra

Abstract:

      Let $\mathcal {A}$ and $\mathcal B$ be unital Banach algebra and $\mathcal M$ be Banach $\mathcal A, \mathcal B$-module. Then $\mathcal T=\big( \begin{smallmatrix} \mathcal {A} & \mathcal M \\ & \mathcal {B} \end{smallmatrix}\big)$ becomes a triangular Banach algebra when equipped with the Banach space norm $\|\big( \begin{smallmatrix} a & m \\ & b \end{smallmatrix} \big)\|=\|a\|_{\mathcal A}+\|m\|_{\mathcal M}+\|b\|_{\mathcal B}$. A Banach algebra $\mathcal T$ is said to be Lie $n$-weakly amenable if all Lie derivations from $\mathcal T$ into its $n^{\text{th}}$ dual space ${\mathcal T}^{(n)}$ are standard. In this paper we investigate Lie $n$-weak amenability of a triangular Banach algebra $\mathcal T$ in relation to that of the algebras $\mathcal A, \mathcal B$ and their action on the module $\mathcal M$.