Preregular maps between Banach lattices

Abstract

A continuous linear map from a Banach lattice E into a Banach lattice F is preregular if it is the difference of positive continuous linear maps from E into the bidual F ″ of F . This paper characterizes Banach lattices B with either of the following properties: (1) for any Banach lattice E , each map in L(E, B) is preregular; (2) for any Banach lattice F , each map in L(B, F) is preregular. It is shown that B satisfies (1) (repectively (2)) if and only if B ′ satisfies (2) (respectively (1)). Several order properties of a Banach lattice satisfying (2) are discussed and it is shown that if B satisfies (2) and if B is also an atomic vector lattice then B is isomorphic as a Banach lattice to 1 1 (Γ) for some index set Γ.