Weak amenability of triangular Banach algebras

Abstract:

Let $\mathcal {A}$ and $\mathcal {B}$ be unital Banach algebras, and let $\mathcal {M}$ be a Banach $\mathcal {A},\mathcal {B}$-module. Then $\mathcal {T} = \begin {bmatrix}\mathcal {A} & \mathcal {M}\\ 0 & \mathcal {B} \end {bmatrix}$ becomes a triangular Banach algebra when equipped with the Banach space norm $\left \Vert \begin {bmatrix} a & m\\ 0 & b \end {bmatrix} \right \Vert = \Vert a \Vert _{\mathcal {A}} + \Vert m \Vert _{\mathcal {M}} + \Vert b \Vert _{\mathcal {B}}$. 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 $\mathcal {M}$.