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Representations of the Dedekind Completions of Spaces of Continuous Functions

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Positivity and Noncommutative Analysis

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

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Abstract

Various representations of the Dedekind completions of Riesz spaces of continuous functions are known in the literature. The aim of this review paper is to collect together some of these results and to connect the various constructions to each other.

Dedicated to Prof. Ben de Pagter, on the occasion of his 65th birthday

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Notes

  1. 1.

    The general construction, which applies to an arbitrary partially ordered set, is due to MacNeille [16].

  2. 2.

    The operators I and S were first introduced by Baire [4] for functions of one real variable. Subsequently, Sendov [19] extended the definition to interval functions, see Sect. 4, and Anguelov [2] considered functions defined on an arbitrary topological space.

  3. 3.

    The relevant results in [23] deal with nearly finite normal semi-continuous functions, but it is evident that these results apply equally in the locally bounded case.

  4. 4.

    If X is not a Baire space, then C d(X) may fail to be a Riesz subspace of B ℓoc(X). In fact, it may not even be a linear subspace of B ℓoc(X).

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Author information

Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa

    Jan Harm van der Walt

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  1. Jan Harm van der Walt
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Correspondence to Jan Harm van der Walt .

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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der Walt, J.H.v. (2019). Representations of the Dedekind Completions of Spaces of Continuous Functions. In: Buskes, G., et al. Positivity and Noncommutative Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-10850-2_27

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