Necessary Conditions for the Inheritance of Weak Amenability in Triangular Banach Algebras

Abstract

In this paper, we investigate the weak amenability of triangular Banach algebras of the form $T = \operatorname{Tri}(A, B, M)$, where $A$ and $B$ are Banach algebras and $M$ is a Banach $(A,B)$-bimodule. Given a closed ideal $J = \operatorname{Tri}(I_A, I_B, N)$ of $T$, we establish sufficient conditions under which the weak amenability of the quotient algebra $T/J$ implies the weak amenability of $T$. Furthermore, we present counterexamples to show that each of these conditions is necessary.