Functional extenders and set-valued retractions

We describe the supports of a class of real-valued maps on $C*(X)$ introduced by Radul. Using this description, a characterization of compact-valued retracts of a given space in terms of functional extenders is obtained. For example, if $X\subset Y$, then there exists a continuous compact-valued retraction from $Y$ onto $X$ if and only if there exists a normed weakly additive extender $u\colon C*(X)\to C*(Y)$ with compact supports preserving $\min$ (resp., $\max$) and weakly preserving $\max$ (resp., $\min$). Similar characterizations are obtained for upper (resp., lower) semi-continuous compact-valued retractions. These results provide characterizations of (not necessarily compact) absolute extensors for zero-dimensional spaces, as well as absolute extensors for one-dimensional spaces, involving non-linear functional extenders.

💡 Research Summary

The paper investigates a class of real‑valued functionals on the Banach space $C^{*}(X)$ of bounded continuous functions on a Tychonoff space $X$, originally introduced by Radul. These functionals are required to satisfy four axioms: (i) normalization ($\mu(1_{X})=1$), (ii) weak additivity ($\mu(f+c_{X})=\mu(f)+c$ for constant $c$), (iii) preservation of the maximum ($\mu(\max{f,g})=\max{\mu(f),\mu(g)}$), and (iv) weak preservation of the minimum ($\mu(\min{f,c_{X}})=\min{\mu(f),c}$). The families of all such functionals are denoted $R_{\max}(X)$ (max‑preserving, min‑weakly‑preserving) and $R_{\min}(X)$ (min‑preserving, max‑weakly‑preserving).

A central technical contribution is a precise description of the support $S(\mu)$ of any functional $\mu\in R_{\max}(X)\cup R_{\min}(X)$. The support is defined as the set of points $x\in\beta X$ (the Čech–Stone compactification) for which the values of $\mu$ can be altered by modifying functions only in arbitrarily small neighbourhoods of $x$. Lemma 2.2 shows that $S(\mu)$ coincides with the intersection of a family $A_{\mu}$ of closed subsets of $\beta X$ that are “indistinguishable” for $\mu$. Moreover, when $\mu$ is weakly additive and monotone, $S(\mu)$ belongs to $A_{\mu}$ and is the smallest element of $A_{\mu}$.

Using a secondary family $\Lambda_{\mu}$ of closed sets, the authors prove that every such functional can be represented as
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