Regular operators from and into a small Riesz space
The authors consider in great detail the spaces \(L^ r\) (resp. \(L^ b\)) of all regular (resp. order bounded) operators from and into the smallest possible non-Dedekind complete Riesz space \(\ell_ 0^ \infty\) of all eventually constant sequences. We confine ourselves to mentioning some of the most striking results: (i) if \(E\) is a uniformly complete Riesz space, then \(L^ r(E,\ell_ 0^ \infty)=L^ b(E,\ell_ 0^ \infty)\); furthermore, \(L^ r(E,\ell_ 0^ \infty)\) is a Riesz space in which all lattice operations may be computed with the Riesz-Kantorovich formula. (ii) \(L^ r(\ell_ 0^ \infty,F)=L^ b(\ell_ 0^ \infty,F)\) for all Archimedean Riesz spaces \(F\); moreover, \(L^ r(\ell_ 0^ \infty,F)\) is a Riesz space if and only if \(F\) is Dedekind \(\sigma\)- complete. Hence, \(L^ r(\ell_ 0^ \infty)\) is not a Riesz space. It even fails to have the Riesz decomposition property. For the many other results we refer to this interesting paper itself.
scientific article; zbMATH DE number 57308
scientific article