Weighted Composition Operators ofC0(X)'s

Abstract

In this paper, we prove that into isometries and disjointness preserving linear Ž . Ž . maps from C X into C Y are essentially weighted composition operators 0 0Tf s h и f ( for some continuous map and some continuous scalar-valued function h. ᮊ

Mappings preserving zero products

“…If we set The following example tells us that it is necessary to work with B * * rather than B for the co-domain of ϕ in the last corollary. In [21,Example 9], one can see that it is also necessary to assume A is unital. In case θ preserves all zero products in A, we have the following To prove Theorem 4.11, we need the following results.…”

“…Then θ is a ring isomorphism if and only if θ(f ) = f • σ for a homeomorphism σ from Y onto X (see, e.g., [17, p. 57]). θ is a surjective isometry if and only if θ(f ) = hf • σ for a homeomorphism σ from Y onto X and a unimodular continuous function h by the Banach-Stone Theorem (see, e.g., [21]). In other words, θ = hϕ, a product of h and an algebra isomorphism ϕ.…”

“…In fact, θ preserves zero products if and only if θ(f ) = hf •σ, where h can be zero somewhere and σ is a general continuous map (see, e.g., [21]). In [1,2,12,16,18,20,21], to name a few, the above relations are extended to the case where θ is not bijective, or is not continuous, or maps between vector-valued function spaces. We are interested in the question if the zero-product preserving property still plays an important role in a general algebraic setting.…”

Disjointness preserving mappings on BSE Ditkin algebras

“…[1], [2] or [6]), on spaces of continuous functions (see e.g. [14], [3], [7], [15] or [12]), on group algebras of locally compact Abelian groups ( [8]), on Fourier algebras ( [10] and [20]) and on some others (see e.g. [16], [17] or [5]).…”

Separating maps on weighted function algebras on topological groups

“…P r o o f. The result follows from Banach-Stone theorem; see for example [7]. Here, we give a direct proof for the sake of completeness.…”