It is shown that the existence of a biseparating map between a large class of spaces of vector-valued continuous functions A(X,E) and A(Y,F) implies that some compactifications of X and Y are homeomorphic. In some cases, conditions are given to warrant the existence of a homeomorphism between the realcompactifications of X and Y; in particular we find remarkable differences with respect to the scalar context: namely, if E and F are infinite-dimensional and T: C*(X,E) → C*(Y,F) is a biseparating map, then the realcompactifications of X and Y are homeomorphic.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm172-1-3,
author = {Jesus Araujo},
title = {Realcompactness and spaces of vector-valued functions},
journal = {Fundamenta Mathematicae},
volume = {173},
year = {2002},
pages = {27-40},
zbl = {0997.46028},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm172-1-3}
}
Jesus Araujo. Realcompactness and spaces of vector-valued functions. Fundamenta Mathematicae, Tome 173 (2002) pp. 27-40. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm172-1-3/