Weak amenability of triangular Banach algebras

Let ${\mathcal A}$ and ${\mathcal B}$be unital Banach algebras, and let ${\mathcal M}$ be a Banach ${\mathcal A},{\mathcal B}$-module. Then ${\mathcal T} = \left[ \begin{array}{cc} {\mathcal A} & {\mathcal M} 0 & {\mathcal B} \end{array} \right]$ becomes a triangular Banach algebra when equipped with the Banach space norm $\ensuremath {\Vert}\left[ \begin{array}{cc} a & m 0 & b \end{array} \right] \... ...rt} _{{\mathcal M}} + \ensuremath {\Vert} b \ensuremath {\Vert} _{{\mathcal B}}$. A Banach algebra ${\mathcal T}$is said to be $n$-weakly amenable if all derivations from ${\mathcal T}$ into its $n^{\mathrm{th}}$ dual space ${\mathcal T}^{(n)}$ are inner. In this paper we investigate Arens regularity and $n$-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 ${\mathcal M}$.