Amalgamated duplication of Banach algebras from homological point of view

In a paper from 2002, Zhang [Weak amenability of module extensions of Banach algebras, Trans. Amer. Math. Soc. 354 (2002) 4131-4151] used module extensions to construct an example of a weakly amenable Banach algebra which is not 3-weakly amenable. Following his approach, we show that for amalgamated duplication of dual Banach algebras, we are able to choose appropriate conditions for weak Connes amenability of those Banach algebras which extend the Zhang's result in a much more general setting. Moreover, apart from the characterization of minimal idempotents, we restrict our investigation to the study of injectivity, projectivity and flatness of amalgamated duplication of Banach algebras. Particularly, it is worth mentioning that our result paves the way for studying of homological properties and weak Connes amenability of those dual Banach algebras that have a semidirect product-like structure including Lau algebras.