# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Charalambos D. Aliprantis and Owen Burkinshaw, Positive operators, Pure and Applied Mathematics, vol. 119, Academic Press, Inc., Orlando, FL, 1985. · Zbl 0608.47039 [2] 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 [3] Z. Ercan and S. Onal, An answer to a conjecture of Cao, Reilly and Xiong, to appear in Czechoslovak Math. J. · Zbl 1164.46311 [4] 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 [5] 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.