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Some remarks on Phelps property U of a Banach space into C(K) spaces

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Abstract

A subspace X of a Banach space Y has Property U whenever every continuous linear functional on X has a unique norm-preserving (i.e., Hahn–Banach) extension to Y (Phelps in Trans Am Math Soc 95:238–255, 1960). Throughout this document, we introduce and develop a systematic study of the existence of U-embeddings between Banach spaces X and Y, that is, isometric embeddings of X into Y whose ranges have property U. In particular, we focus on the case \(Y=C(K)\), where K is a compact Hausdorff topological space. We provide results for X a reflexive or another C(K) space.

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Acknowledgements

The authors were partially supported by the Universitat Politècnica de València (Spain) and by Grant PID2021-122126NB-C33 funded by national MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”. Ch. Cobollo was also supported by Generalitat Valenciana (Project PROMETEU/2021/070 and CIACIF/2021/378) and by MCIN/AEI/ 10.13039/ 501100011033 (Project PID2019-105011GB). A. J. Guirao was also supported by Fundación Séneca, Región de Murcia (Grant 19368/PI/14). V. Montesinos was also supported by AEI/FEDER (Project MTM2017-83262-C2-1-P of Ministerio de Economía y Competitividad). We thank two anonymous referees for their valuable suggestions to improve the content and structure of the present paper. In particular, Remark 4.10 and Proposition 6.31 were suggested by one of them.

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  1. Universitat Politècnica de València, Valencia, Comunitat Valenciana, Spain

    Christian Cobollo, Antonio José Guirao & Vicente Montesinos

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  1. Christian Cobollo
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  2. Antonio José Guirao
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Cobollo, C., Guirao, A.J. & Montesinos, V. Some remarks on Phelps property U of a Banach space into C(K) spaces. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 6 (2024). https://doi.org/10.1007/s13398-023-01504-9

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