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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 {\it E. Beckenstein} and {\it L. Narici} [J. Math. Anal. Appl. 192, No. 1, 258-265 (1995; Zbl 0828.47024)] and with {\it 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 $\Bbb{K}\cup \{ \infty \}$. 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.

##### MSC:
 46E40 Spaces of vector- and operator-valued functions 54D60 Realcompactness and real compactification (general topology) 54C35 Function spaces (general topology) 54C40 Algebraic properties of function spaces (general topology)
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