Injective and projective Banach modules

Abstract

In this paper we extend Semadeni's definition [8] of a free, a projective and of an injective Banach space to the case of Banach modules. As in the algebraic case a projective module is a generalization of a free module, and its dual is an injective module, that means, has the “extension property”. Free, projective and injective Banach modules are studied following a line which has some resemblances with Northcott's procedure in [5]. In this connection see also Rodriguez [7]. It is shown that every module is a quotient of a projective module and every module can be embedded in a “unique smallest” injective module, which is called an injective envelope (Th. 2.17, Th. 3.18). The last section is devoted to L 1(G) -modules.