Banach-Stone theorem for Banach lattice valued continuous functions

Abstract

Abstract. Let X and Y be compact Hausdorff spaces, E be a Banach lattice and F be an AM space with unit. Let π : C(X, E) → C(Y, F ) be a Riesz isomorphism such that 0 ∈ f (X) if and only if 0 ∈ π(f )(Y ) for each f ∈ C(X, E). We prove that X is homeomorphic to Y and E is Riesz isomorphic to F . This generalizes some known results.

Riesz isomorphisms of tensor products of order unit Banach spaces

“…The following theorem (Theorem 2.2) extends the main result of [6]. In order to prove this we recall the Banach-Stone theorem for A(K) spaces, when K is a simplex, due to Lazar [8].…”

“…Also when K is a Choquet simplex with the set of extreme points ∂ e K, closed (the so-called Bauer simplex), A(K) is isometric (via the restriction map) to C(∂ e K). Thus it is interesting to consider questions similar to those answered in [6] for the family of Choquet simplexes and order unit Banach spaces. We recall that S = {e * ∈ E * : e * (e) = 1 = e * } is called the state space of E. An extreme convex set F ⊂ K is called a face.…”

“…Let π : C(X, E) → C(Y, F ) be a Riesz isomorphism (i.e., orderpreserving linear bijection) such that 0 / ∈ f (X) if and only if 0 / ∈ π(f )(Y ) for each f ∈ C(X, E). Ercan andÖnal have proved in [6] that E is Riesz isomorphic to F and X is homeomorphic to Y . This has been extended to the case of Banach lattices in [5].…”

The uniform separation property and Banach–Stone theorems for lattice-valued Lipschitz functions

“…We can cite, among others, the generalizations obtained by J. X. Chen, Z. L. Chen and N.-C. Wong [5], Z. Ercan and S.Önal [6,7] and X. Miao, J. Cao and H. Xiong [12].…”

Common zeros preserving maps on vector-valued function spaces and Banach modules

“…For some results concerning non-vanishing preserving maps (on lattices) one can see [4,6,5,8,9,15,18,19,20]. We also refer to [14] for some relevant concepts in the case of scalar-valued continuous functions.…”