Generalized injectivity of Banach modules

TL;DR

This paper investigates the concept of 0-injectivity in Banach modules, specifically characterizing it for group algebra modules and demonstrating its presence in certain dual and L^p spaces.

Contribution

It provides a characterization of 0-injectivity for L^1(G) modules and shows that L^1(G)^{**} and L^p(G) are 0-injective Banach modules.

Findings

L^{1}(G) is 0-injective as a left module.

L^{1}(G)^{**} is 0-injective.

L^{p}(G) for 1<p<∞ is 0-injective.

Abstract

In this paper, we study the notion of $\phi$-injectivity in the special case that $\phi=0$. For an arbitrary locally compact group $G$, we characterize the 0-injectivity of $L^{1}(G)$ as a left $L^{1}(G)$ module. Also, we show that $L^{1}(G)^{**}$ and $L^{p}(G)$ for $1<p<\infty$ are 0-injective Banach $L^{1}(G)$ modules.