On barycentrically soft compacta

It is shown that a barycentrically soft compactum is necessarily an absolute retract of weight ≤ ω1. Since softness of a map is the mapping version of the property of a space to be an absolute retract, the above mentioned result can be considered as mapping version of the Ditor–Haydon Theorem stating that if P (X) is an absolute retract then the compactum X is of weight ≤ ω1 [2]. All spaces considered are assumed to be compacta (compact Hausdorff spaces). For a compactum X let C(X) be the space of all real-valued continuous functions on X metrized by sup-metric and let P (X) be the space of all non-negative functionals μ : C(X) → R with norm 1, equipped with the weak* topology. Recall that the base of the weak* topology in P (X) consists of the sets of the form O(μ0, f1, . . . , fn, ε) = {μ ∈ P (X) | |μ(fi)−μ0(fi)| < ε for every 1 ≤ i ≤ n}. Let E be a locally convex vector space. Then for any convex compact subset K ⊂ E there exists a map b = bK : P (X) → K which is called the barycentric map of probability measures. It is defined by b(μ) = ∫ x dμ(x), where x = idE . The map bK is continuous [1]. It is not difficult to check that for μ = a1δx1 + . . .+ anδxn , ai ∈ R, xi ∈ K, we have bK(μ) = a1 x1 + . . .+ an xn, where δxi denotes the Dirac measure supported by xi. A map f : X → Y is said to be (0-)soft if for any (0-dimensional) paracompact space Z, any closed subspace A of Z and maps Φ : A → X and Ψ : Z → Y with Ψ |A = f ◦ Φ there exists a map G : Z → X such that G|A = Φ and Ψ = f ◦G. This notion was introduced by E. Shchepin [9]. 1991 Mathematics Subject Classification: 46E27, 54C10, 54C55.