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Geometry of Spaces of Orthogonally Additive Polynomials on C(K)

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Abstract

We study the space of orthogonally additive n-homogeneous polynomials on C(K). There are two natural norms on this space. First, there is the usual supremum norm of uniform convergence on the closed unit ball. As every orthogonally additive n-homogeneous polynomial is regular with respect to the Banach lattice structure, there is also the regular norm. These norms are equivalent, but have significantly different geometric properties. We characterise the extreme points of the unit ball for both norms, with different results for even and odd degrees. As an application, we prove a Banach–Stone theorem. We conclude with a classification of the exposed points.

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Acknowledgements

We thank Dirk Werner and Tony Wickstead for useful discussions. We would also like to thank the referee for a careful reading of the paper and helpful suggestions.

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Authors and Affiliations

  1. School of Mathematics & Statistics, University College Dublin, Belfield, Dublin 4, Ireland

    Christopher Boyd

  2. School of Mathematics, Statistics and Applied Mathematics, National University of Ireland Galway, Galway, Ireland

    Raymond A. Ryan & Nina Snigireva

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  1. Christopher Boyd
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  2. Raymond A. Ryan
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  3. Nina Snigireva
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Correspondence to Raymond A. Ryan.

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Boyd, C., Ryan, R.A. & Snigireva, N. Geometry of Spaces of Orthogonally Additive Polynomials on C(K). J Geom Anal 30, 4211–4239 (2020). https://doi.org/10.1007/s12220-019-00240-0

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