On Non-Archimedean Subadditive Separating Maps by Edward Beckenstein and Lawrence Narici

Electronic Version 24 Summer (1999)

On Non-Archimedean Subadditive Separating Maps

Edward Beckenstein and Lawrence Narici

Let X and Y denote compact Tihonov spaces. Let C(X) and C(Y) denote the spaces of continuous functions on X and Y, respectively, taking values in a non-Archimedean, nontrivially valued field (K, | |). A map H:C(X) --> C(Y) is separating if f(x) g(x) = 0 for all x \in X implies that Hf(y) Hg(y) = 0 for all y \in Y. If f(x) g(x) = 0 for all x \in X if and only if Hf(y) Hg(y) = 0 for all y \in Y, we say that H is biseparating. Results about automatic continuity and the form of additive and linear separating maps have been developed in [ABN1, ABN2, ABN3, ABN4, BN1, BNT, HBN, NBA]. Under certain circumstances, additive separating maps induce a homeomorphism h:Y --> X and are of the form Hf(y) = H(f(h(y))1)(y); maps of this form are called pseudocomposition maps. In this article we investigate subadditive separating maps. We show that a bijective, biseparating subadditive map H must be a pseudocomposition map; in addition, such maps are ``norm bounded''. We present a number of examples to demonstrate the necessity of certain hypotheses.

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