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A Survey on Isometries Between Lipschitz Spaces

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Associative and Non-Associative Algebras and Applications (MAMAA 2018)

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Abstract

The famous Banach–Stone theorem, which characterizes surjective linear isometries between C(X) spaces as certain weighted composition operators, has motivated the study of isometries defined on different function spaces (see [33, 34]). The research on surjective linear isometries between spaces of Lipschitz functions is a subject of long tradition which goes back to the sixties with the works of de Leeuw [61] and Roy [81], and followed by those by Mayer-Wolf [67], Weaver [97], Araujo and Dubarbie [3], and Botelho, Fleming and Jamison [8]. This topic continues to attract the attention of some authors (see [44, 52, 62]). In the setting of Lipschitz spaces, we present a survey on non-necessarily surjective linear isometries and codimension 1 linear isometries [55], vector-valued linear isometries [56], local isometries and generalized bi-circular projections [54], 2-local isometries [52, 57], projections and averages of isometries [12] and hermitian operators [13, 14]. We also raise some open problems on bilinear isometries and approximate isometries in the same context.

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Acknowledgements

The authors wish to thank the referee for making several suggestions which improved this paper.

The first author’s research was partially supported by Predoctoral contract for the Personnel Research Training 2016 of University of Almería, and the second author’s one by the Spanish Ministry of Economy and Competitiveness project no. MTM2014-58984-P and the European Regional Development Fund (ERDF), and Junta of Andalucía grant FQM-194.

Some of the results of this survey were presented in the talks “Advances about isometries on Lipschitz spaces” by A. Jiménez–Vargas and “Bilinear isometries on algebras of Lipschitz functions” by Moisés Villegas-Vallecillos at The 3rd Moroccan Andalusian Meeting on Algebras and their Applications (Chefchaouen, Morocco, April 12–14, 2018). They would like to thank the organizers for the hospitality during their stay.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Almería, 04120, Almería, Spain

    M. G. Cabrera-Padilla & A. Jiménez-Vargas

  2. Departamento de Matemáticas, Universidad de Cádiz, 11510, Puerto Real, Spain

    Moisés Villegas-Vallecillos

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  1. M. G. Cabrera-Padilla
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  2. A. Jiménez-Vargas
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  3. Moisés Villegas-Vallecillos
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Correspondence to A. Jiménez-Vargas .

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Editors and Affiliations

  1. Departamento Álgebra, Geometría y Topología, Universidad de Málaga, Málaga, Spain

    Mercedes Siles Molina

  2. Department of Algebra, University of Granada, Ceuta, Spain

    Laiachi El Kaoutit

  3. Department of Mathematics, Abdelmalek Essaâdi University, Tetouan, Morocco

    Mohamed Louzari

  4. Department of Mathematics, Abdelmalek Essaâdi University, Tetouan, Morocco

    L'Moufadal Ben Yakoub

  5. Department of Mathematics, Abdelmalek Essaâdi University, Tetouan, Morocco

    Mohamed Benslimane

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Cabrera-Padilla, M.G., Jiménez-Vargas, A., Villegas-Vallecillos, M. (2020). A Survey on Isometries Between Lipschitz Spaces. In: Siles Molina, M., El Kaoutit, L., Louzari, M., Ben Yakoub, L., Benslimane, M. (eds) Associative and Non-Associative Algebras and Applications. MAMAA 2018. Springer Proceedings in Mathematics & Statistics, vol 311. Springer, Cham. https://doi.org/10.1007/978-3-030-35256-1_3

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