Memorandum on multiplicative bijections and order

Date

2009-08-01

Author

Cabello Sanchez, Felix
Cabello Sanchez, Javier
ERCAN, ZAFER
Önal, Süleyman

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This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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Let C(X, I) denote the semigroup of continuous functions from the topological space X to I = [0, 1], equipped with the pointwise multiplication. The paper studies semigroup homomorphisms C(Y, I) -> C(X, I), with emphasis on isomorphisms. The crucial observation is that, in this setting, homomorphisms preserve order, so isomorphisms preserve order in both directions and they are automatically lattice isomorphisms. Applications to uniformly continuous and Lipschitz functions on metric spaces are given. Sample result: if Y and X are complete metric spaces of finite diameter without isolated points, every multiplicative bijection T : Lip(Y, I) -> Lip(X, I) has the form Tf = f circle tau, where tau : X -> Y is a Lipschitz homeomorphism.

Subject Keywords

Algebra and Number Theory

URI

https://hdl.handle.net/11511/35250

Collections

Department of Mathematics, Article