On the Banach–Stone problem

Abstract

Abstract. Let X and Y be locally compact Hausdorff spaces, let E and F be Banach spaces, and let T be a linear isometry from C 0 (X, E) into C 0 (Y, F ). We provide three new answers to the Banach-Stone problem: (1) T can always be written as a generalized weighted composition operator if and only if F is strictly convex; (2) if T is onto then T can be written as a weighted composition operator in a weak sense; and (3) if T is onto and F does not contain a copy of ∞ 2 then T can be written as a weighted composition operator in the classical sense.

Extreme Point Methods and Banach-Stone Theorems

“…[ 12] PROOF. Given t e ch(N) and y* e N*(t) there exists s e ch(M) and x* e ext(X(s)*) such that T*(y* o \jr,) = x* o \j/ s .…”

“…Jeang and Wong [12] have observed that if E is real and not strictly convex, there exist norm-one elements u u u 2 in E such that Then the operator T from C({1, 2}, K) into C({1, 2}, E) by is an isometry. The next two examples show that the conclusion of Theorem 3.4 and Theorem 3.5 can fail if either the density of the linear span of N*(t) or the triviality of Z(E 2 (0) is not satisfied.…”

“…A theorem of Cambern [6] shows that if the Banach space Y is strictly convex, then any isometry T as above is a generalized weighted composition operator. Moreover, it has been pointed out by Jeang and Wong [12] that, at least in the case of real spaces, if every isometry T from C 0 (Q, X) into Co(K, Y) is a generalized weighted composition operator for any Q, X, K, then Y must be strictly convex.…”

Biseparating linear maps between continuous vector-valued function spaces

“…After Jerison [23], many authors worked on different variants of the vector-valued Banach-Stone Theorem (see [12,16,18,20,[22][23][24]). In particular, as an extension of the representation theorem of Abramovich [1], Hernandez, Beckenstein and Narici proved in [12] that if T is an isometric biseparating linear map from C(X, E) onto C(Y, F) then T is a weighted composition operator Tf (y) -h{y)(f {<p{y))).…”

Finite codimensional isometries on spaces of vector-valued continuous functions

“…It is easy to see that the map T : C(X, E) −→ C(Y, F ) defined, for each f ∈ C(X, E), by (T f )(x, i) := f (x), e i (where {e 1 , e 2 } is the canonical basis in K 2 ), and (T f )(p j ) := 0 for all j, is a linear isometry with codimension n 0 . As in [17], it can be checked that T is not a weighted composition map.…”