Functional extenders and set-valued retractions

We describe the supports of a class of real-valued maps on C∗(X) introduced by Radul (2009) [6]. Using this description, a characterization of compact-valued retracts of a given space in terms of functional extenders is obtained. For example, if X⊂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:C∗(X)→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.