On a question on Banach - Stone theorem

We present a very simple and elementary proof of the main theorem of [l]. This also gives an answer to a conjecture in [1].

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 n : C ( X, E ) ^ C(Y ) is a Riesz isomorphism such that n(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 G Y be fixed and П у : E ^ R be defined by n y () = n (1 0 e )( y ), where 1 0 e(x) = e . It is obvious that n y 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