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Zbl 0918.46026
Araujo, Jesús
Separating maps and linear isometries between some spaces of continuous functions.
(English)
[J] J. Math. Anal. Appl. 226, No.1, 23-39 (1998). ISSN 0022-247X

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.
[L.Janos (Kent/Ohio)]
MSC 2000:
*46E15 Banach spaces of functions defined by smoothness properties
46B04 Isometric theory of Banach spaces

Keywords: locally compact Hausdorff space; additive map; existence of a biseparating map; homeomorphic

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