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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 π:C(X,E)C(Y×N) be a Riesz isomorphism such that 0f(X)0π(f)(Y×N) for all fC(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 π:C(X×M)C(Y×N) be a unit preserving Riesz isomorphism. If 0f({x}×M) for all xX 0πf({y}×N) for all yY, then X,M are homeomorphic to Y,N, respectively.

46E40Spaces of vector- and operator-valued functions
46B42Banach lattices