When each continuous operator is regular, II

1996/06/12 by Abramovich, Yuri A., Wickstead, Anthony W.
#46B42 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.math/9606210

Abstract

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), (⋆) 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 (⋆). 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 (⋆). 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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