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Lattice Homomorphisms in Harmonic Analysis

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Positivity and Noncommutative Analysis

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

Abstract

Let S be a non-empty, closed subspace of a locally compact group G that is a subsemigroup of G. Suppose that X, Y , and Z are Banach lattices that are vector sublattices of the order dual \(\mbox{C}_{\mbox{c}}(S,\mathbb R)^\sim \) of the real-valued, continuous functions with compact support on S, and where Z is Dedekind complete. Suppose that ∗ : X × Y  → Z is a positive bilinear map such that supp (x ∗ y) ⊆ supp x ⋅ supp y for all x ∈ X+ and y ∈ Y+ with compact support. We show that, under mild conditions, the canonically associated map from X into the vector lattice of regular operators from Y  into Z is then a lattice homomorphism. Applications of this result are given in the context of convolutions, answering questions previously posed in the literature.

As a preparation, we show that the order dual of the continuous, compactly supported functions on a closed subspace of a locally compact space can be canonically viewed as an order ideal of the order dual of the continuous, compactly supported functions on the larger space.

As another preparation, we show that Lp-spaces and Banach lattices of measures on a locally compact space can be embedded as vector sublattices of the order dual of the continuous, compactly supported functions on that space.

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

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Acknowledgements

The results in this article were obtained in part when the first author held the Kloosterman Chair in Leiden in October 2017, and when the second author visited Lancaster University in October 2018. The financial support by the Mathematical Institute of Leiden University and the London Mathematical Society is gratefully acknowledged.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Lancaster, Lancaster, UK

    H. Garth Dales

  2. Mathematical Institute, Leiden University, Leiden, The Netherlands

    Marcel de Jeu

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

    Marcel de Jeu

Authors
  1. H. Garth Dales
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  2. Marcel de Jeu
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Corresponding author

Correspondence to Marcel de Jeu .

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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Dales, H.G., Jeu, M.d. (2019). Lattice Homomorphisms in Harmonic Analysis. In: Buskes, G., et al. Positivity and Noncommutative Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-10850-2_6

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