Functional extenders and set-valued retractions

Abstract

We describe the supports of a class of real-valued maps on $C\ast (X)$ 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 $u:C\ast (X)\to C\ast (Y)$ 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.

19 pages

References in corpus (1)

• Spaces of idempotent measures of compact metric spaces