Extensions and liftings of positive linear mappings on Banach lattices

Abstract

Let F be a closed sublattice of a Banach lattice G. We show that any positive linear mapping from F into L 1 ( μ ) {L^1}(\mu ) or C ( X ) C(X) for a Stonian space X has a positive norm preserving extension to G. A dual result for positive norm preserving liftings is also established. These results are applied to obtain extension and lifting theorems for order summable and majorizing linear mappings. We also obtain some partial results concerning positive extensions and liftings of compact linear mappings.