The Jordan decomposition and half-norms

Abstract

Let % be a Banach space, with norm 11 * II, ordered by a positive cone + and order the dual ** by the dual cone $* . We prove that, if Q> is orthogonally generated, each/ G % * has an orthogonal, and norm-unique, Jordan decomposition / = /+ -/_ with/ G *, if, and only if, the norm on % has the order theoretic property |||| = inf{>0; -w < a < t? for some u 9 v e,}, when $] is the unit ball of . Various characterizations of the canonical half-norm associated with $ + are also given.