GENERALIZED INJECTIVITY OF BANACH MODULES

Abstract. In this paper, we study the notion of φ-injectivity in the special case that φ = 0. For an arbitrary locally compact group G, we characterize the 0-injectivity of L1(G) as a left L1(G) module. Also, we show that L1(G)∗ ∗ and Lp(G) for 1 < p < ∞ are 0-injective Banach L1(G) modules. 1. introduction The homological properties of Banach modules such as injectivity, pro-jectivity, and flatness were first introduced and investigated by Helemskii; see [5, 6]. White in [11] gave a quantitative version of these concepts, i.e., he introduced the concepts of C-injective, C-projective, and C-flat Banach modules for a positive real number C. Recently Nasr-Isfahani and Soltani Renani introduced a version of these homological concepts based on a char-acter of a Banach algebra A and they showed that every injective (projective, flat) Banach module is a character injective (character projective, character flat respectively) module but that the converse is not valid in general. With the use of these new homological concepts, they gave a new characterization of φ-amenability of a Banach algebra A such that φ ∈ ∆(A) and a necessary condition for φ-contractibility of A; see [8]. 2. preliminaries Let A be a Banach algebra and ∆(A) denote the character space of A, i.e., the space of all non-zero homomorphisms from A onto C. We denote by A-mod and mod-A the category of all Banach left A-modules and all Banach right A-modules, respectively. In the case that A has an identity we denote by A-unmod the category of all Banach left unital modules. For E,F ∈ A-mod, let AB(E,F) be the space of all bounded linear left A-module morphisms from E into F