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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
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