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Lipschitz algebras and peripherally-multiplicative maps

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Abstract

Let X be a compact metric space and let Lip(X) be the Banach algebra of all scalarvalued Lipschitz functions on X, endowed with a natural norm. For each f ∈ Lip(X), σ π (f) denotes the peripheral spectrum of f. We state that any map Φ from Lip(X) onto Lip(Y) which preserves multiplicatively the peripheral spectrum:

$$ \sigma _\pi (\Phi (f)\Phi (g)) = \sigma _\pi (fg),\forall f,g \in Lip(X), $$

is a weighted composition operator of the form Φ(f) = τ · (f ○ φ) for all f ∈ Lip(X), where τ: Y → {−1, 1} is a Lipschitz function and φ: Y → X is a Lipschitz homeomorphism. As a consequence of this result, any multiplicatively spectrum-preserving surjective map between Lip(X)-algebras is of the form above.

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Authors and Affiliations

  1. Departamento de Álgebra y Análisis Matemático, Universidad de Almería, 04071, Almería, Spain

    A. Jiménez-Vargas & Moisés Villegas-Vallecillos

Authors
  1. A. Jiménez-Vargas
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  2. Moisés Villegas-Vallecillos
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Corresponding author

Correspondence to A. Jiménez-Vargas.

Additional information

This work is partly supported by MEC project MTM2006-4837, and Junta de Andalucía projects P06-FQM-1215 and P06-FQM-1438

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Jiménez-Vargas, A., Villegas-Vallecillos, M. Lipschitz algebras and peripherally-multiplicative maps. Acta. Math. Sin.-English Ser. 24, 1233–1242 (2008). https://doi.org/10.1007/s10114-008-7202-4

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  • DOI: https://doi.org/10.1007/s10114-008-7202-4

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