Banach-Stone theorems for vector-valued functions
by
Denny H. Leung
National University of Singapore
Coauthors: Wee-Kee Tang

Let X and Y be completely regular spaces and E and F be Hausdorff topological vector spaces. We call a linear map T from a subspace of C(X, E) into C(Y, F) a Banach-Stone map if it has the form Tf(y) = S_y(f(h(y)) for a family of linear operators S_y : E →F, y ∈ Y and a function h: Y → X. In this talk, we consider maps having the property:

(Z)   ∩ki=1Z(fi) ≠ ∅⇔ ∩ki=1Z(Tfi) ≠ ∅,

(1)

where Z(f) = {f = 0}. We characterize linear bijections with property (Z) between spaces of continuous functions, respectively, spaces of differentiable functions, as Banach-Stone maps. In particular, we have the following theorem.

Theorem. Suppose that X and Y are realcompact spaces and E and F are Hausdorff topological vector lattices (respectively, C^*-algebras). Let T: C(X, E) → C(Y, F) be a vector lattice isomorphism (respectively, *-algebra isomorphism) such that

Z(f) ≠ ∅⇔ Z(Tf) ≠ ∅.

Then X is homeomorphic to Y and E is lattice isomorphic (respectively, C^*-isomorphic) to F.

Some results concerning the automatic continuity of T are also obtained.

Date received: February 26, 2008