Non-Transferability of Quotient Weak Amenability in Banach Algebras

Abstract

We study a variant of weak amenability in Banach algebras, called $\frac{A}{‎J}‎$-weak amenability, where derivations take values in the dual of a quotient algebra. While weak amenability relative to an ideal has been previously studied, little is known about its behavior when comparing nested ideals. In this paper, we examine how $\frac{A}{J}‎$‎-weak amenability behaves as the ideal $J$ varies. We provide examples and counterexamples showing that this property does not generally transfer between ideals, even when the algebra itself is weakly amenable. Our findings underline the importance of the ideal's position and show that additional conditions are often required to preserve amenability properties in quotient settings.