Biseparating linear maps between continuous vector-valued function spaces

Abstract

LetA\ y be compact Hausdorff spaces and £, F be Banach spaces. Alinearmap T :is separating if Tf, Tg have disjoint cozeroes whenever / , g have disjoint cozeroes. We prove that a biseparating linear bijection T (that is, T and T~l are separating) is a weighted composition operator Tf = hf o<p. Here, h is a function from Y into the set of inveitible linear operators from E onto F, and <p is a homeomorphism from Y onto X. We also show that T is bounded if and only if h{y) is a bounded operator from E onto F for all y in Y. In this case, h is continuous with respect to the strong operator topology.

Disjointness preserving maps between vector-valued group algebras

“…The following example, adapted from [10], shows that, in the vector-valued setting, unlike in the complex-valued case ( [9]), the automatic continuity of T cannot be obtained from its disjointness preserving property, even if T is a biseparating bijection.…”

“…(see, for example, [1,2,3,4,5,7,8,13,16]). In recent years, certain attention has been given to such maps when defined on spaces of vector-valued continuous functions (see, e.g., [10,14]). However, we do not know much about disjointness preserving maps on vectorvalued settings in comparison with scalar-valued contexts and something similar can be said with regard to (algebra) homomorphisms between vector-valued group algebras.…”

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

“…(ii) The following example borrowed from [11,Example 2.4] shows that in the above results when E (and so F ) is not a finite dimensional space, common zeros preserving maps need not be continuous (see also [1,Remarks(2)…”

“…In the case where X, Y are metric spaces and E, F are normed spaces, a complete description of linear bijections T : A(X, E) → A(Y, F ) between certain subspaces of C(X, E) and C(Y, F ), satisfying the weaker condition Z(f ) ∩ Z(g) = ∅ ⇐⇒ Z(T f ) ∩ Z(T g) = ∅, for all f, g ∈ A(X, E), is given in [7] and then some extensions of the previous results are obtained. In [18], among other things, the authors considered maps preserving zero set containments, which dates back to [11]. Indeed, they characterized linear bijections T : C(X, E) → C(Y, F ) such that…”

Nonlinear order isomorphisms on function spaces

“…Later advances saw versions of the theorem of Gelfand and Kolmogorov on other function algebras as well as generalizations to biseparating maps, i.e., bijective maps that preserve disjointness of functions in both directions. For instance, see [3,4,5,15,18,20,22]. As a vector lattice, Kaplansky [21] showed that C(X) determines a compact Hausdorff space X up to homeomorphism.…”