Riesz completions of some spaces of regular operators
We describe the Riesz completion (in the sense of van Haandel) of some spaces of regular operators as explicitly identified subspaces of the regular operators into larger range spaces
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arXiv subjects
Anthony W. Wickstead
Publications and source records attributed to Anthony W. Wickstead.
Tensor Products of Archimedean Riesz Spaces: a Representational Approach
A representational approach to constructing the Fremlin tensor product of two Archimedean Riesz spaces. [Warning: do not view the HTML version!]
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When each continuous operator is regular, II
The following theorem is essentially due to L.~Kantorovich and B. Vulikh and it describes one of the most important classes of Banach lattices between which each continuous operator is regular. {\bf Theorem 1.1.} {\sl Let $E$ be an arbitrary L-space and $F$ be an arbitrary Banach lattice with Levi norm. Then ${\cal L}(E,F)={\cal L}^r(E,F),\ (\star) $ that is, every continuous operator from $E$ to $F$ is regular.} In spite of the importance of this theorem it has not yet been determined to what extent the Levi condition is essential for the validity of equality $(\star)$. Our main aim in this work is to prove a converse to this theorem by showing that for a Dedekind complete $F$ the Levi condition is necessary for the validity of $(\star)$. As a sample of other results we mention the following. {\bf Theorem~3.6.} {\sl For a Banach lattice $F$ the following are equivalent: {\rm (a)} $F$ is Dedekind complete; {\rm (b)} For all Banach lattices $E$, the space ${\cal L}^r(E,F)$ is a Dedekind complete vector lattice; {\rm (c)} For all L-spaces $E$, the space ${\cal L}^r(E,F)$ is a vector lattice.}
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