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Banach-Stone theorem for Banach lattice valued continuous functions. (English) Zbl 1127.46026
Let $$X,Y,N$$ be compact Hausdorff spaces and $$E$$ a Banach lattice. Let $$\pi:C(X,E) \rightarrow C(Y\times N)$$ be a Riesz isomorphism such that $$0 \notin f(X)\Leftrightarrow 0 \notin \pi(f)(Y\times N)$$ for all $$f \in C(X,E)$$. The authors show that the above conditions imply that $$E$$ is Riesz homeomorphic to $$C(N)$$ and $$X$$ is homeomorphic to $$Y$$. When $$E$$ is the real scalars, this result was proved by J. Cao, I. Reilly and H. Xiong [Acta Math. Hung. 98, 103–110 (2003; Zbl 1027.46025)].
A key ingredient in the proof is a lemma about Riesz isomorphisms on spaces of continuous functions on products of compact spaces. Let $$\pi:C(X \times M) \rightarrow C(Y \times N)$$ be a unit preserving Riesz isomorphism. If $$0 \notin f(\{x\} \times M)$$ for all $$x \in X$$ $$\Leftrightarrow$$ $$0 \notin \pi f(\{y\} \times N)$$ for all $$y \in Y$$, then $$X,M$$ are homeomorphic to $$Y,N$$, respectively.

##### MSC:
 46E40 Spaces of vector- and operator-valued functions 46B42 Banach lattices
##### Keywords:
Riesz isomorphism; Banach lattice; Banach-Stone theorem
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##### References:
  Charalambos D. Aliprantis and Owen Burkinshaw, Positive operators, Pure and Applied Mathematics, vol. 119, Academic Press, Inc., Orlando, FL, 1985. · Zbl 0608.47039  J. Cao, I. Reilly, and H. Xiong, A lattice-valued Banach-Stone theorem, Acta Math. Hungar. 98 (2003), no. 1-2, 103 – 110. · Zbl 1027.46025  Z. Ercan and S. Onal, An answer to a conjecture of Cao, Reilly and Xiong, to appear in Czechoslovak Math. J. · Zbl 1164.46311  J. J. Font and S. Hernández, On separating maps between locally compact spaces, Arch. Math. (Basel) 63 (1994), no. 2, 158 – 165. · Zbl 0805.46049  E. de Jonge and A. C. M. van Rooij, Introduction to Riesz spaces, Mathematisch Centrum, Amsterdam, 1977. Mathematical Centre Tracts, No. 78. · Zbl 0421.46001
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