Multiplicative semigroups of Lipschitz functions

Autores: Félix Cabello Sánchez , Javier Cabello Sánchez
Localización: Extracta mathematicae, ISSN-e 0213-8743, Vol. 25, Nº 3, 2010 (Ejemplar dedicado a: Proceedings of the Seventh Italian-Spanish Conference of General Topology and its Applications, Badajoz (Spain), September 7-10, 2010), págs. 239-247
Idioma: inglés
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Resumen
- Given a (complete) metric space X, we denote by Lip(X) the space of real-valued Lipschitz functions on X and we equip it with the pointwise product. The purpose of this note is to describe those bijections T : Lip(Y ) �¨ Lip(X) which are \multiplicative " in the sense that whenever f; g �¸ Lip(Y ) are such that fg �¸ Lip(Y ) one has T(fg) = T(f)T(g).
  
  The main result of the paper states that if X has no isolated points, then every mul- tiplicative bijection T : Lip(Y ) �¨ Lip(X) arises as T(f) = f . , where : X �¨ Y is a Lipschitz homeomorphism and so it is automatically linear.
  
  We also give a description of the semigroup isomo