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The Banach–Stone theorem revisited. (English) Zbl 1166.46309
Summary: Let $$X$$ and $$Y$$ be compact Hausdorff spaces, and $$E$$ and $$F$$ be locally solid Riesz spaces. If $$\pi:C(X,E)\rightarrow C(Y,F)$$ is a 1-biseparating Riesz isomorphism then $$X$$ and $$Y$$ are homeomorphic, and $$E$$ and $$F$$ are Riesz isomorphic. This generalizes the main results of Z. Ercan and S. Önal [Proc. Am. Math. Soc. 135, No. 9, 2827–2829 (2007; Zbl 1127.46026)] and X.–H. Miao, J.–L. Cao and H.–Y. Xiong [J. Math. Anal. Appl. 313, No. 1, 177–183 (2006; Zbl 1102.46017)], and answers a conjecture in [Z. Ercan and S. Önal, loc. cit.].

##### MSC:
 46E40 Spaces of vector- and operator-valued functions 46B42 Banach lattices
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##### References:
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