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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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