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.

MSC:

 46E15 Banach spaces of continuous, differentiable or analytic functions 46B04 Isometric theory of Banach spaces
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References:

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