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 \(\beta X\). In \(\beta X\) the author defines the equivalence relation \(\sim\), defined as \(x\sim y\) whenever\(f^{\beta X}(x)=f^{\beta X}(y)\) for every \(f\in A\). In this way, the quotient space \(\gamma X:=\beta X/\sim\) is obtained. The space \(\gamma X\) is a compactification of \(X\) (the Samuelcompactification of \(X\) associated by the uniformity generated by \(A\)), and every \(f\in A\) can be continuously extended to a map from \(\gamma X\) into \(\mathbb{K}\cup \{ \infty \}\). The symbolism \(C(X,E)\) (resp. \(C^{\ast}(X,E)\)) denotes the space of continuous (resp. and bounded) functions on \(X\) taking values in \(E\). Suppose that \(A(X,E)\subset C(X,E)\) is an \(A\)-module, where \(A\) is a subring of \(X\) which separates each point of \(X\) from each point of \(\beta X\). The author says that \(A(X,E)\) is compatible with \(A\) if, there exists \(f\in A(X,E)\) with \(f(x)\not= 0\), and if, given any points \(x,y\in \beta X\) with \(x\sim y\), we have \(\|f\|^{\beta X}(x)=|f\|^{\beta X}(y)\) for every \(f\in A(X,E)\). A subring \(A\subset C(X)\) is said to be strongly regular if given \(x_{0}\in \gamma X\) and a nonempty closed subset \(K\) of \(\gamma X\) which does not contain \(x_{0}\), there exists \(f\in A\) such that \(f^{\gamma X}\equiv 1\) on a neigbourhood of \(x_{0}\) and \(f^{\gamma X}(K)\equiv 0\). A map \(T:A(X,E)\rightarrow A(Y,F)\) is said to be separating if it is additive and \(\|(Tf)(y)\|\cdot \|(Tg)(y)\|=0\) for all \(y\in Y\) whenever \(f,g\in A(X,E)\) satisfy \(\|f(x)\|\cdot \|g(x)\|=0\) for all \(x\in X\). 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 \(A(X,E)\subset C(X,E)\) and \(A(Y,F)\subset C(Y,F)\) are an \(A\)-module and a \(B\)-module compatible with \(A\) and \(B\), respectively, where \(A\subset C(X)\) and \(B\subset C(Y)\) are strongly regular rings. Also, in case when \(\gamma X \not= \beta X\) and \(\gamma Y \not= \beta Y\), it is assumed that for every \(x\in \beta X\) and \(y\in \beta Y\), there exists \(f\in A(X,E)\) and \(g\in A(Y,F)\) satisfying \(\|f\|^{\beta X}(x)\not= 0\) and\(\|g\|^{\beta Y}(y)\not= 0\). If \(T:A(X,E)\rightarrow A(Y,F)\) is a biseparating map, then \(\gamma X\) and \(\gamma Y\) are homeomorphic.(2) If \(T:C^{\ast}(X,E)\rightarrow C^{\ast}(Y,F)\) is biseparating, then \(\upsilon X\) and \(\upsilon Y\) are homeomorphic.


46E40 Spaces of vector- and operator-valued functions
54D60 Realcompactness and realcompactification
54C35 Function spaces in general topology
54C40 Algebraic properties of function spaces in general topology
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