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A geometric Jordan decomposition theorem

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

For a compact convex set K,  let A(K) denote the space of real-valued affine continuous functions, equipped with the supremum norm. For a closed subspace \(X \subset A(K)\) we give sufficient conditions, so that the weak\(^*\) closure of the set of extreme points of the dual unit ball has a decomposition in terms of ‘positive’ and ‘negative’ parts. We give several applications of these ideas to convexity and positivity. When K is a Choquet simplex, we show that the dual unit ball of such an X,  inherits nice facial structure. We also use this to partly solve the open problem of exhibiting faces that are Choquet simplexes in the dual unit ball of a Banach space.

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Acknowledgements

Thanks are due to Professor M. López-Pellicer for his encouragement. Thanks are also due to the referees for their suggestions to improve the exposition. This work is part of the project ‘Classification of Banach spaces using differentiability’, funded by Science and Engineering Research Board (SERB), Core Research Grant, CRG2023-000595. Ideas of this article are being extended to tensor product spaces and spaces of operators, which is an ongoing research committed in this project and will be published elsewhere.

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  1. Department of Mathematics, Shiv Nadar Institution of Eminence, Gautam Buddha Nagar, Greater Noida, UP, 201314, India

    T. S. S. R. K. Rao

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  1. T. S. S. R. K. Rao
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Correspondence to T. S. S. R. K. Rao.

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To the memory of Å. Lima.

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Rao, T.S.S.R.K. A geometric Jordan decomposition theorem. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 68 (2024). https://doi.org/10.1007/s13398-024-01569-0

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  • DOI: https://doi.org/10.1007/s13398-024-01569-0

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