Linear operators with compact supports, probability measures and Milyutin maps

The notion of a regular operator with compact supports between function spaces is introduced. On that base we obtain a characterization of absolute extensors for zero-dimensional spaces in terms of regular extension operators having compact supports. Milyutin maps are also considered and it is established that some topological properties, like paracompactness, metrizability and k-metrizability, are preserved under Milyutin maps.

💡 Research Summary

The paper introduces a new notion of “compact support” for linear operators acting between spaces of continuous (or bounded continuous) functions, and uses this concept to obtain several deep results linking functional analysis, probability measures, and topology.

First, for Tychonoff spaces X and Y and a locally convex linear space E, the authors consider linear maps u:C(X,E)→C(Y,E) (or the bounded version C*(X,E)→C*(Y,E)). For each y∈Y they define an evaluation functional T(y) by T(y)(h)=u(h)(y). The support s(T(y)) is a closed subset of the Čech–Stone compactification βX consisting of points x such that every neighbourhood of x contains a function h with (βh)(βX∖U)=0 but T(y)(h)≠0. The operator u is said to have compact supports if s(T(y))⊂X for every y. Regularity means that T(y)(h) always lies in the closed convex hull of h(X). This regular‑compact‑support condition is equivalent to the existence of a continuous map T:Y→Pc(X,E) (or P*c) where Pc(X,E) denotes the space of regular linear functionals on C(X,E) with compact support, equipped with the pointwise convergence topology. Moreover, when y∈X the measure T(y) reduces to the Dirac measure δy.

Using this framework the authors give a new characterization of absolute extensors for zero‑dimensional spaces (AE(0)). They prove that a space X is AE(0) if and only if for every C‑embedding i:X↪Y there exists a regular extension operator u:C(X)→C(Y) (or C*‑version) having compact supports; equivalently, for any locally convex space E there is a regular extension u:C*(X,E)→C*(Y,E) with compact supports. Thus the classical Dugundji‑AE(0) equivalence is reformulated in terms of linear operators whose supports stay inside X.

The paper then studies the functors Pc and Pc. Pc(X) consists of all regular linear maps µ:C(X)→ℝ with compact support; Pc(X) is the analogous space for bounded functions. Both are endowed with the pointwise convergence topology, making them covariant functors on the category of Tychonoff spaces. When X is realcompact, Pc(X) is homeomorphic to the closed convex hull of the canonical embedding eX:X→RC(X) (the space of real‑valued continuous functions with the product topology). Consequently Pc(X) is metrizable exactly when X is compact and metrizable. The map jX:Pc(X)→P*c(X) (restriction to bounded functions) is a homeomorphism precisely for pseudo‑compact X; otherwise it fails to be a topological isomorphism.

Milyutin maps are defined as surjections f:X→Y admitting a regular averaging operator u:C(X)→C(Y) with compact supports, equivalently a continuous map T:Y→Pc(X) such that the support of T(y) is contained in the fibre f⁻¹(y). The authors show that every metric space Y is the image of a 0‑dimensional metric space X under a perfect Milyutin map, and that every p‑paracompact space is the image of a 0‑dimensional p‑paracompact space via a perfect Milyutin map.

Finally, the paper proves that several important topological properties are preserved under Milyutin maps: paracompactness, collectionwise normality, (complete) metrizability, stratifiability, δ‑metrizability, and k‑metrizability. In particular, they answer positively a question of Shchepin by showing that every AE(0) space is k‑metrizable (Corollary 5.5).

Overall, the work provides a unified perspective that connects linear extension operators with compact supports, probability measure functors, and Milyutin maps, yielding new characterizations of AE(0) spaces and demonstrating the robustness of many topological properties under these maps.