On a Question on Banach–stone Theorem

In this paper we use the standard terminology and notations of the Riesz spaces theory (see [2]). The Banach lattice of the continuous functions from a compact Hausdorff space into a Banach lattice E is denoted by C(K,E). If E = R then we write C(K) instead of C(K,E). 1 stands for the unit function in C(K). One version of the Banach–Stone theorem states that: Theorem 1. Let X and Y be compact Hausdorff spaces. Then C(X) and C(Y ) are Riesz isomorphic if and only if X and Y are homeomorphic. An elementary proof of this theorem can be found in [2]. This theorem is generalized in [l] as follows. Theorem 2. Let X and Y be compact Hausdorff spaces and E be a Banach lattice. If π : C(X,E)→ C(Y ) is a Riesz isomorphism such that π(f) has no zeros whenever f has no zero, then X and Y are homeomorphic and E is Riesz isomorphic to R. A quite difficult and long proof of the previous theorem is given without using Theorem 1 in [2] and it is conjectured that Theorem 2 follows from Theorem 1. In this paper we give an answer to this conjecture with an elementary proof as follows. C Proof of Theorem 2. Clearly E is nonzero. Let ∈ Y be fixed and πy : E → R be defined by πy() = π(1 ⊗ e)(y), where 1 ⊗ e(x) = e. It is obvious that πy is one-to-one and Riesz homomorphism. So, E is Riesz isomorphic onto a nonzero Riesz subspace of R, As E is nonzero and dimension of R is one, E is Riesz isomorphic to R. This complete the proof and answers to the conjecture in [1]. B