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Separating maps and linear isometries between some spaces of continuous functions. (English) Zbl 0918.46026
For a given locally compact Hausdorff space $X$, a Banach space $E$ and a function $\sigma: X\to (0,\infty)$ satisfying certain conditions, the author defines the Banach space $C^\sigma_0(X,E)$ of continuous functions from $X$ into $E$. An additive map $T: C^\sigma_0(X, E)\to C^\tau_0(Y, F)$ between two such Banach spaces is said to be separating if whenever $f,g\in C^\sigma_0(X,E)$ satisfy $\| f(x)\| \| g(x)\|= 0$ for every $x\in X$, then $\|(Tf)(y)\| \|(Tg)(y)\|= 0$ for every $y\in Y$. $T$ is said to be biseparating if it is bijective and both $T$ and $T^{-1}$ are separating. The author proves that the existence of a biseparating map $T: C^\sigma(X, E)\to C^\tau_0(Y, F)$ implies that the spaces $X$ and $Y$ are homeomorphic.

46E15Banach spaces of continuous, differentiable or analytic functions
46B04Isometric theory of Banach spaces
Full Text: DOI
[1] J. Araujo, Biseparating maps, automatic continuity, and Banach--Stone maps · Zbl 1056.46032
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