On separating subadditive maps

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_Ycolon 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.