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 (i¿) or 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.The main purpose of this paper is to establish some conditions under which positive linear mappings between Banach lattices have norm preserving positive linear extensions or liftings.These results are then applied to obtain extension and lifting theorems for order summable and majorizing linear mappings and to establish the inductive and projective character of the |cr|tensor product topology introduced by Jacobs [6].Finally, we obtain some partial results concerning the problem of finding positive, compact linear extensions and liftings of positive compact linear mappings between Banach lattices.We begin our discussion with a summary of known results concerning the corresponding problems for continuous linear mappings between Banach spaces.Suppose that E, F and G are Banach spaces, that cf> is an isometry of F into G and that T is a continuous linear mapping of F into E.When does T have a norm preserving linear extension T:G -*E, that is, when does there exist a continuous linear mapping T of G into E such that ||T|| = ||T|| and such that the following diagram commutes?If E is the scalar field, the Hahn-Banach theorem asserts that such an extension T of T always exists.Results of Nachbin [lO], Kelley [9] and Goodner