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.