Extremal Structure and Duality of Lipschitz Free Spaces
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We analyse the relationship between different extremal notions in Lipschitz free spaces (strongly exposed, exposed, preserved extreme and extreme points). We prove in particular that every preserved extreme point of the unit ball is also a denting point. We also show in some particular cases that every extreme point is a molecule, and that a molecule is extreme whenever the two points, say x and y, which define it satisfy that the metric segment [x, y] only contains x and y. The most notable among them is the case when the free space admits an isometric predual with some additional properties. As an application, we get some new consequences about norm attainment in spaces of vector-valued Lipschitz functions.
KeywordsExtreme point dentability Lipschitz free duality uniformly discrete
Mathematics Subject ClassificationPrimary 46B20 Secondary 54E50
The authors are grateful to Ramón Aliaga and Antonio Guirao for sending them their preprint. The first and the last authors are grateful to the Laboratoire de Mathématiques de Besançon for the excellent working conditions during their visit in June 2017. The third author is grateful to Departamento de Análisis Matemático of Universidad de Granada for hospitality and excellent working conditions during his visit in June 2017. The authors would like to thank Ginés López-Pérez and Matías Raja for useful conversations. Finally, the authors are very grateful to the anonymous referee for their suggestions and their careful reading of the paper.
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