Lipschitz free p -spaces for 0 < p < 1
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LIPSCHITZ FREE p-SPACES FOR 0 < p < 1 BY
Fernando Albiac∗ Mathematics Department–InaMat, Universidad P´ ublica de Navarra Campus de Arrosad´ıa, Pamplona 31006, Spain e-mail: [email protected] AND
Jos´ e L. Ansorena∗∗ Department of Mathematics and Computer Sciences, Universidad de La Rioja Logro˜ no 26004, Spain e-mail: [email protected] AND
´th† Marek Cu Faculty of Mathematics and Physics, Department of Mathematical Analysis Charles University, 186 75 Praha 8, Czech Republic e-mail: [email protected]ff.cuni.cz AND
Michal Doucha†† Institute of Mathematics, Czech Academy of Sciences 115 67 Praha 1, Czech Republic e-mail: [email protected] ∗ F. Albiac acknowledges the support of the Spanish Ministry for Economy and
Competitivity Grants MTM2014-53009-P for An´ alisis 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 An´ alisis Vectorial, Multilineal y Aplicaciones. † M. C´ uth has been supported by Charles University Research program ˇ 17-04197Y. No. UNCE/SCI/023 and by the Research grant GACR †† M. Doucha was supported by the GACR ˇ project 16-34860L and RVO: 67985840. Received March 18, 2019 and in revised form April 15, 2019
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F. ALBIAC ET AL.
Isr. J. Math.
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) Fp (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 p for all 0 < p ≤ 1. On the other hand, solving a problem by the first author and N. Kalton, there are metric spaces N ⊂ M such that the natural embedding from Fp (N ) to Fp (M) is not an isometry.
1. Introduction It is safe to say that most of the research in functional analysis is done in the framework of Banach spaces. While the theory of the geometry of these spaces has evolved very rapidly over the past sixty years, by contrast, the study of the more general case of quasi-Banach spaces has lagged far behind despite the fact that the first papers in the subject appeared in the early 1940’s ([4, 7]). The neglect of non-locally convex spaces within functional analysis is easily understood. Even when they are complete and metrizable, working with them requires doing without one of the most powerful tools in Banach spaces: the Hahn–Banach theorem and the duality techniques that rely on it. This difficulty in even making the simplest initial steps has led some to regard quasi-Banach spaces as too challenging and consequently they have
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