The Jordan decomposition and half-norms

Abstract

Let B be a Banach space, with norm ‖ · ‖, ordered by a positive cone B+ and order the dual B* by the dual cone B+*. We prove that, if B is orthogonally generated, each f ∈ B* has an orthogonal, and norm-unique, Jordan decomposition f = f+ - f− with f± ∈ B*, if, and only if, the norm on B has the order theoretic property, when B1 is the unit ball of B. Various characterizations of the canonical half-norm associated with B+ are also given. Show more