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Realcompactness and spaces of vector-valued functions. (English) Zbl 0997.46028
This paper is dedicated to describe the linear biseparating maps defined between spaces of vector-valued continuous functions on realcompact spaces. There are thereby generalized to the vector-valued setting several results previously proved by the author, with E. Beckenstein and L. Narici [J. Math. Anal. Appl. 192, No. 1, 258-265 (1995; Zbl 0828.47024)] and with J. J. Font [Proc. Edinb. Math. Soc., II. Ser. 43, No. 1, 139-147 (2000; Zbl 0945.46032)], for biseparating maps defined between spaces of scalar-valued functions. Assume that X is a completely regular Hausdorff space and A is subring of C(X) which separates each point of X from each point of βX. In βX the author defines the equivalence relation , defined as xy wheneverf βX (x)=f βX (y) for every fA. In this way, the quotient space γX:=βX/ is obtained. The space γX is a compactification of X (the Samuelcompactification of X associated by the uniformity generated by A), and every fA can be continuously extended to a map from γX into 𝕂{}. The symbolism C(X,E) (resp. C * (X,E)) denotes the space of continuous (resp. and bounded) functions on X taking values in E. Suppose that A(X,E)C(X,E) is an A-module, where A is a subring of X which separates each point of X from each point of βX. The author says that A(X,E) is compatible with A if, there exists fA(X,E) with f(x)0, and if, given any points x,yβX with xy, we have f βX (x)=|f βX (y) for every fA(X,E). A subring AC(X) is said to be strongly regular if given x 0 γX and a nonempty closed subset K of γX which does not contain x 0 , there exists fA such that f γX 1 on a neigbourhood of x 0 and f γX (K)0. A map T:A(X,E)A(Y,F) is said to be separating if it is additive and (Tf)(y)·(Tg)(y)=0 for all yY whenever f,gA(X,E) satisfy f(x)·g(x)=0 for all xX. Moreover, T is said to be biseparating if it is bijective and both T and T -1 are separating. Among others, the following main results are obtained: (1) Suppose that A(X,E)C(X,E) and A(Y,F)C(Y,F) are an A-module and a B-module compatible with A and B, respectively, where AC(X) and BC(Y) are strongly regular rings. Also, in case when γXβX and γYβY, it is assumed that for every xβX and yβY, there exists fA(X,E) and gA(Y,F) satisfying f βX (x)0 andg βY (y)0. If T:A(X,E)A(Y,F) is a biseparating map, then γX and γY are homeomorphic.(2) If T:C * (X,E)C * (Y,F) is biseparating, then υX and υY are homeomorphic.
MSC:
46E40Spaces of vector- and operator-valued functions
54D60Realcompactness and real compactification (general topology)
54C35Function spaces (general topology)
54C40Algebraic properties of function spaces (general topology)