Generalized injectivity of Banach modules (Q2786764)

Let \(A\) be a Banach algebra. A Banach left \(A\)-module is injective if, for all Banach left \(A\)-modules \(E\) and \(F\), and every admissible bounded left \(A\)-module monomorphism \(T : F\to E\), the induced mapping \(T_J:{}_A\mathcal{B}(E,J)\to {}_A\mathcal{B}(F,J)\), defined by NEWLINE\[NEWLINE T_J(R) = R \circ T\quad\text{for each bounded left \(A\)-module mapping \(R:E\to J\),} NEWLINE\]NEWLINE is onto. This is a well-known concept in the Banach algebra cohomology theory. In this paper, the author introduces an extension of this to a notion of \(0\)-injectivity, by requiring that the above condition is only checked for those monomorphisms \(T\) with the additional property that \(\text{Im\,}T\) contains \(A\cdot E\). Basic properties of this new notion are proved, as well as examples (from the standard Banach left \(L^1(G)\)-modules such as \(L^p(G)\), for locally compact groups \(G\)) are considered.