Functional extenders and set-valued retractions

TL;DR

This paper characterizes compact-valued retractions and absolute extensors in topological spaces using functional extenders, linking set-valued retractions with properties of certain real-valued maps on function spaces.

Contribution

It introduces a new description of supports of Radul's maps and provides characterizations of retractions and absolute extensors via non-linear functional extenders.

Findings

Characterization of compact-valued retractions using functional extenders.

Conditions for the existence of continuous compact-valued retractions.

Characterizations of absolute extensors for zero- and one-dimensional spaces.

Abstract

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.