A relationship between the space of orthomorphisms and the centre of a vector lattice
The authors consider an arbitrary vector lattice \(\mathbb{Z}\). Recall that the centre \(\mathbb{Z}(E)\) of \(E\) consists of all linear operators on \(E\) which are bounded by some multiple of the identity. Consider the collection \(\mathrm{Orth}(E)\) of all orthomorphisms on \(E\). The authors show that \(\mathbb{Z}(\mathrm{Orth\,} E)\) and \(\mathbb{Z}(E)\) are isomorphic, and then they give a necessary and sufficient condition for the equality \(\mathrm{Orth}(E)= \mathbb{Z}(E)\). It is given in terms of Banach lattices.
scientific article; zbMATH DE number 6484278
scientific article