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Linear and Multilinear Isometries in a Noncompact Framework

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Abstract

Both classical linear and multilinear isometries defined between subalgebras of bounded continuous functions on (complete) metric spaces are studied. Particularly, we prove that certain such subalgebras, including the subalgebras of uniformly continuous, Lipschitz or locally Lipschitz functions, determine the topology of (complete) metric spaces. As a consequence, it is proved that the subalgebra of Lipschitz functions determines the Lipschitz in the small structure of a complete metric space. Furthermore, we provide a weighted composition representation for multilinear isometries from similar subalgebras on (not necessarily complete) metric spaces. We apply this general representation to obtain more specific ones for subalgebras of uniformly continuous and Lipschitz functions.

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Authors and Affiliations

  1. Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, 16315-1618, Iran

    Maliheh Hosseini

  2. Departamento de Matemáticas, Universitat Jaume I, Campus Riu Sec, 8029 AP, Castellón, Spain

    Juan J. Font

Authors
  1. Maliheh Hosseini
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  2. Juan J. Font
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Correspondence to Maliheh Hosseini.

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J. J. Font was supported by Spanish Government Grant PID2019-106529GB-I00 and by Universitat Jaume I (Projecte UJI-B2019-08).

This work was partially supported by a grant from the IMU-CDC.

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Hosseini, M., Font, J.J. Linear and Multilinear Isometries in a Noncompact Framework. Mediterr. J. Math. 18, 86 (2021). https://doi.org/10.1007/s00009-021-01737-1

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  • DOI: https://doi.org/10.1007/s00009-021-01737-1

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