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When Do the Regular Operators Between Two Banach Lattices Form a Lattice?

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Positivity and Noncommutative Analysis

Part of the book series: Trends in Mathematics ((TM))

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Abstract

We investigate when the regular operators from one Banach lattice into another form a Riesz space. We give complete results when the domain is either separable or has an order continuous norm. In these two settings, at least, the lattice operators are given by the Riesz-Kantorovich formulae, in contrast with Elliott’s negative result for the general setting.

This work is dedicated to Ben de Pagter on the occasions of his 65th birthday and his impending retirement

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Notes

  1. 1.

    This result, essentially, was proved independently around 1936 by each of Freudenthal [8], Kantorovich [9], and F. Riesz [12]. In spite of the historical justification for naming the Riesz-Kantorovich formulae after all three of these authors, we stick to what is by now the accepted terminology.

  2. 2.

    This actually characterizes atomic Banach lattices with an order continuous norm, which is due to van Rooij, see [13] and [14].

  3. 3.

    After this paper was completed, Michael Elliott has informed the author that he can prove the existence of an atomic AM-space with property (⋆) which does not have an order continuous norm.

  4. 4.

    These functions are simply, apart from numbering, an uncountable analogue of the usual Rademacher functions.

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Authors and Affiliations

  1. Mathematical Sciences Research Centre, Queen’s University Belfast, Belfast, Northern Ireland

    A. W. Wickstead

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  1. A. W. Wickstead
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Correspondence to A. W. Wickstead .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Mississippi, Oxford, MS, USA

    Gerard Buskes

  2. Mathematical Institute, Leiden University, Leiden, The Netherlands; Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa

    Marcel de Jeu

  3. College of Science and Engineering, Flinders University, Adelaide, Australia

    Peter Dodds

  4. Department of Mathematics, University of South Carolina, Columbia, SC, USA

    Anton Schep

  5. School of Mathematics and Statistics, University of New South Wales, Sydney, Australia

    Fedor Sukochev

  6. Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands

    Jan van Neerven

  7. Mathematical Sciences Research Centre, Queen’s University Belfast, Belfast, UK

    Anthony Wickstead

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Wickstead, A.W. (2019). When Do the Regular Operators Between Two Banach Lattices Form a Lattice?. In: Buskes, G., et al. Positivity and Noncommutative Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-10850-2_29

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