Weak module amenability of triangular Banach algebras

Let A and B be unital Banach algebras and let M be a unital Banach A,B-module. Forrest and Marcoux 6] have studied the weak amenability of triangular Banach algebra \(\mathcal{T} = \left {_B^{AM} } \right]\) and showed that T is weakly amenable if and only if the corner algebras A and B are weakly amenable. When \(\mathfrak{A}\) is a Banach algebra and A and B are Banach \(\mathfrak{A}\)-module with compatible actions, and M is a commutative left Banach \(\mathfrak{A}\)-A-module and right Banach \(\mathfrak{A}\)-B-module, we show that A and B are weakly \(\mathfrak{A}\)-module amenable if and only if triangular Banach algebra T is weakly \(\mathfrak{T}\)-module amenable, where \(\mathfrak{T}: = \{ ^\alpha _\alpha ]:\alpha \in \mathfrak{A}\} \).