Biprojectivity and Biflatness of Bi-amalgamated Banach Algebras

Abstract

Let A, X be two Banach algebras and let X be an algebraic Banach A-module equipped with a bounded bilinear map $$\Theta :X\times X\rightarrow A$$ which is compatible with the A-module operations of X. Then the $$\ell ^1$$ -direct sum $$A\times X$$ endowed with the multiplication $$\begin{aligned} (a,x)(b,y)=(ab+\Theta (x,y),ay+xb+xy) \quad (a,b\in A, x, y\in X) \end{aligned}$$ is a Banach algebra, denoted by $$A\boxtimes _\Theta X$$ and will be called a bi-amalgamated Banach algebra. Many known Banach algebras such as (generalized) module extension Banach algebras, Lau product Banach algebras, generalized matrix Banach algebras have this general framework. The main aim of this paper is to investigate biprojectivity and biflatness of $$A\boxtimes _\Theta X$$ . Our results extend several results in the literature and provide simple direct proofs for some known results. In particular, we characterize the biprojectivity and biflatness of certain classes of the module extension Banach algebras and generalized matrix Banach algebras. Some unsolved questions are also included.