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Turkish Journal of Mathematics

VESKO VALOV

DOI

10.3906/mat-1401-12

Abstract

Recall that a map T \colon C(X,E) \to C(Y,F), where X, Y are Tychonoff spaces and E, F are normed spaces, is said to be separating, if for any 2 functions f,g \in C(X,E) we have c(T(f)) \cap c(T(g))= \varnothing provided c(f) \cap c(g) = \varnothing. Here c(f) is the co-zero set of f. A typical result generalizing the Banach--Stone theorem is of the following type (established by Araujo): if T is bijective and additive such that both T and T^{-1} are separating, then the realcompactification \nu X of X is homeomorphic to \nu Y. In this paper we show that a similar result is true if additivity is replaced by subadditivity (a map T is called subadditive if T(f+g)(y) \leq T(f)(y) + T(g)(y) for any f,g \in C(X,E) and any y \in Y). Here is our main result (a stronger version is actually established): if T \colon C(X,E) \to C(Y,F) is a separating subadditive map, then there exists a continuous map S_Y\colon \beta Y \rightarrow \beta X. Moreover, S_Y is surjective provided T(f)=0 iff f=0. In particular, when T is a bijection such that both T and T^{-1} are separating and subadditive, \beta X is homeomorphic to \beta Y. We also provide an example of a biseparating subadditive map from C(R) onto C(R), which is not additive.

Keywords

Function spaces, separating maps, supports, subadditive maps

First Page

168

Last Page

173

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