A Triple of Infinite Iterates of the Functor of Positively Homogeneous Functionals

When is the space of semi-additive functionals an absolute (neighbourhood) retract?

“…The construction OS determines a functor acting in the category Comp of compact Hausdorff spaces and their continuous maps. Moreover, the functor OS : Comp Ñ Comp is normal [4]. The notion of normality of functors one can find in [6].…”

“…The following statement shows that the formula (1.2) gives a general form of functionals from OS(X): for every ν P OS(X) there exists a non-empty convex compact set A in P (X) such that ν = ν A , and for each φ P C(X) there exists µ P A such that ν A (φ) = µ(φ), [4].…”

“…It is clear that µ F is a weakly additive, order-preserving, normed, positivelyhomogeneous and semi-additive functional. Moreover, the correspondence F Þ Ñ µ F is one-to-one, [4]. Therefore, we can identify the set F with the functional µ F , and thus exp X Ă OS(X).…”

“…For a compact Hausdorff space X the spaces OS(X) and cc(P (X)) are homeomorphic [4]. Namely, a homeomorphism τ : cc(P (X)) Ñ OS(X) may be given by the rule τ (A) = ν A , A P cc(P (X)).…”

“…Note that OS ω (X) is everywhere dense in OS(X). For every semi-additive functional µ with finite support there exist, see [4], a unique convex closed set A P cc(P (X)), such that…”