Ultrasubadditive Separating Maps
by
Edward Beckenstein
St. John's University
Coauthors: Lawrence Narici (St. John's University)

Let C( X) and C( Y) denote the spaces of R- or C-valued functions on the Tihonov spaces X and Y. 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. When 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. If H is a linear, bijective, biseparating map, then the realcompactifications of X and Y are homeomorphic; if X and Y are realcompact, then H is a weighted composition map, a map of the form Hf( y) = w( y) f( h( y)) for any f in C( X) , and y in Y where h:Y --> X is a homeomorphism; if X and Y are compact and H is only additive, then H is of the form Hf( y) = H[f( h(y) ) 1]( y) , f in C( X) , y in Y, where 1 is the function identically equal to 1. In this article we investigate what happens when the continuous functions take values in a non-Archimedean valued field and H is an ultrasubadditive separating map, i.e., H is separating and | H(f+g) ( x) | <= max( | Hf( x)| , | Hg( x) | ) for all f, g in C(X) and x in X.

Date received: June 22, 1999