Weak amenability of triangular Banach algebras

Abstract

Let A \mathcal {A} and B \mathcal {B} be unital Banach algebras, and let M \mathcal {M} be a Banach A , B \mathcal {A},\mathcal {B} -module. Then T = [ A a m p ; M 0 a m p ; B ] \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 ‖ [ a a m p ; m 0 a m p ; b ] ‖ = ‖ a ‖ A + ‖ m ‖ M + ‖ b ‖ B \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 T \mathcal {T} is said to be n n -weakly amenable if all derivations from T \mathcal {T} into its n t h n^{\mathrm {th}} dual space T ( n ) \mathcal {T}^{(n)} are inner. In this paper we investigate Arens regularity and n n -weak amenability of a triangular Banach algebra T \mathcal {T} in relation to that of the algebras A \mathcal {A} , B \mathcal {B} and their action on the module M \mathcal {M} .