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Lipschitz free p-spaces for 0 < p < 1

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Abstract

This paper initiates the study of the structure of a new class of p-Banach spaces, 0 <p < 1, namely the Lipschitz free p-spaces (alternatively called Arens—Eells p-spaces) \(\mathcal{F}_{p}(\mathcal{M})\) over p-metric spaces. We systematically develop the theory and show that some results hold as in the case of p = 1, while some new interesting phenomena appear in the case 0 <p < 1 which have no analogue in the classical setting. For the former, we, e.g., show that the Lipschitz free p-space over a separable ultrametric space is isomorphic to ℓp for all 0 <p ≤ 1. On the other hand, solving a problem by the first author and N. Kalton, there are metric spaces \(\mathcal{N}\subset\mathcal{M}\) such that the natural embedding from \(\mathcal{F}_{p}(\mathcal{N})\) to \(\mathcal{F}_{p}(\mathcal{M})\) is not an isometry.

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Acknowledgment

F. Albiac and J. L. Ansorena would like to thank the Faculty of Mathematics and Physics at Charles University in Prague for their hospitality and generosity during their visit in September 2018, when most of the work on this paper was undertaken.

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Authors and Affiliations

  1. Mathematics Department-InaMat, Universidad Pública de Navarra, Campus de Arrosadía, Pamplona, 31006, Spain

    Fernando Albiac

  2. Department of Mathematics and Computer Sciences, Universidad de La Rioja, Logroño, 26004, Spain

    José L. Ansorena

  3. Faculty of Mathematics and Physics, Department of Mathematical Analysis, Charles University, 186 75, Praha 8, Czech Republic

    Marek Cúth

  4. Institute of Mathematics, Czech Academy of Sciences, 115 67, Praha 1, Czech Republic

    Michal Doucha

Authors
  1. Fernando Albiac
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  2. José L. Ansorena
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  3. Marek Cúth
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  4. Michal Doucha
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Corresponding author

Correspondence to Fernando Albiac.

Additional information

F. Albiac acknowledges the support of the Spanish Ministry for Economy and Competitivity Grants MTM2014-53009-P for Análisis Vectorial, Multilineal y Aplicaciones, and MTM2016-76808-P for Operators, lattices, and structure of Banach spaces as well as the Spanish Ministry for Science and Innovation under Grant PID2019-1077701GB-I00.

J. L. Ansorena acknowledges the support of the Spanish Ministry for Economy and Competitivity Grant MTM2014-53009-P for Análisis Vectorial, Multilineal y Aplicaciones.

M. Cúth has been supported by Charles University Research program No. UNCE/SCI/023 and by the Research grant GACR 17-04197Y.

M. Doucha was supported by the GAČR project 16-34860L and RVO: 67985840.

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Albiac, F., Ansorena, J.L., Cúth, M. et al. Lipschitz free p-spaces for 0 < p < 1. Isr. J. Math. 240, 65–98 (2020). https://doi.org/10.1007/s11856-020-2061-5

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  • DOI: https://doi.org/10.1007/s11856-020-2061-5

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