Hereditary properties of character injectivity with applications to semigroup algebras

TL;DR

This paper explores the hereditary properties of $$-injectivity in Banach modules over semigroup algebras, revealing new insights into module injectivity related to characters and providing examples of $$-injective modules that are not injective.

Contribution

It introduces and studies hereditary properties of $$-injectivity for Banach modules, especially in the context of semigroup algebras, and presents examples illustrating these concepts.

Findings

Hereditary properties of $$-injectivity for Banach modules are established.

$$-injectivity is analyzed in the context of $ ext{l}^1$-semilattice algebras.

An example of a non-injective but $$-injective Banach module is provided.

Abstract

In this paper, we investigate the notion $\phi$-injectivity for Banach $A$-modules, where $\phi$ is a character on $A.$ We obtain some hereditary properties of $\phi$-injectivity for certain classes of Banach modules related to closed ideals. These results allow us to study $\phi$-injectivity of certain Banach $A$-modules in commutative case, specially $\ell^{1}$-semilattice algebras. As an application, we give an example of a non-injective Banach module which is $\phi$-injective for each character $\phi.$