Biprojectivity of generalized module extension and second dual of Banach algebras

Abstract

In this paper, we study biprojectivity of generalized module extension Banach algebras and second dual of Banach algebras. As a main result, we prove that if [Formula: see text] is a contractible closed ideal of [Formula: see text] such that [Formula: see text] is biprojective, then [Formula: see text] is biprojective. As a consequence, we give some results on biprojectivity of generalized module extension Banach algebras. Indeed, this is a generalization of results in [A. Ebaddian and A. Jabbari, Biprojectivity and biflatness of amalgamated duplication of Banach algebras, J. Algebra Its Appl. 19(7) (2020) 2050132 (15 pp.)]. Moreover, we show that if [Formula: see text] is biprojective and [Formula: see text] is a two-sided ideal of [Formula: see text] then [Formula: see text] is biprojective. This fact generalizes some known results in [M. S. Moslehian and A. Niknam, Biflatness and biprojectivity of second dual of Banach algebras, Southeast Asian Bull. Math 27(1) (2003) 129–133].