## Hereditary properties of character injectivity with applications to semigroup algebras.(English)Zbl 1329.46046

Summary: 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 $$\phi$$.

### MSC:

 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 43A20 $$L^1$$-algebras on groups, semigroups, etc. 46M10 Projective and injective objects in functional analysis
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