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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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