Disjointness preserving and local operators on algebras of differentiable functions

Abstract

This article is to discuss the automatic continuity properties and the representation of disjointness preserving linear mappings on certain normal Fre´chet algebras of complex-valued functions. This class of operators is defined by the condition that any pair of functions with disjoint cozero sets is mapped to functions with disjoint cozero sets, and subsumes the class of local operators. It turns out that such operators are always continuous outside some finite singularity set of the underlying topological space. Our main emphasis is on disjointness preserving operators from Fre´chet algebras of differentiable functions. Such operators are shown to admit a canonical representation that involves weighted composition for the derivatives. This result extends the classical characterization of local operators as linear partial differential operators.1991 Mathematics Subject Classification. Primary 47B38. Secondary 47B48, 46H40, 46J15.

Linear Orthogonality Preservers of Hilbert Bundles

“…For example, a theorem of Peetre [19] states that local linear maps of the space of smooth functions defined on a manifold modeled on R n are exactly the linear differential operators (see [18]). This was extended to the case of vector-valued differentiable functions defined on a finite-dimensional manifold by Kantrowitz and Neumann [14] and Araujo [3], and to the Banach C 1…”

Zero product preserving maps on C1[0,1]

“…(12) see [5,Lemma 1]. The purpose of this section is to consider the condition (12) for an operator T from C 1 [0, 1] into a left Banach C 1 [0, 1]-module X .…”

“…Anyway, we shall keep the terminology from [2] which is more standard in algebraic and noncommutative setting. For historic comments and references about these operators we refer the reader to [2] and [5].…”

“…Let us refer the reader to the paper by Kantrowitz and Neumann [5] which considers a zero product preserving continuous linear operator T which is defined on C m (Ω), where m 0 and Ω is an open subset of R n , and maps into C (Γ ) where Γ is a locally compact Hausdorff space. Although there are some differences in the general setting, Kantrowitz and Neumann have treated in [5] a more general problem and Corollary 4.4 is not really surprising in view of their achievements. Yet this corollary is of some interest because of its proof which does not use standard tools of the theory of zero product preserving operators between function algebras such as supporting points.…”

Biseparating maps between Lipschitz function spaces

“…Separating maps, also called disjointness preserving maps, between spaces of scalar-valued continuous functions defined on compact or locally compact spaces have drawn the attention of researchers in last years (see for instance [10,15,17,21]). …”