The cone of functionals on the Cuntz semigroup

The functionals on an ordered semigroup S in the category Cu--a category to which the Cuntz semigroup of a C*-algebra naturally belongs--are investigated. After appending a new axiom to the category Cu, it is shown that the "realification" S_R of S has the same functionals as S and, moreover, is recovered functorially from the cone of functionals of S. Furthermore, if S has a weak Riesz decomposition property, then S_R has refinement and interpolation properties which imply that the cone of functionals on S is a complete distributive lattice. These results apply to the Cuntz semigroup of a C*-algebra. At the level of C*-algebras, the operation of realification is matched by tensoring with a certain stably projectionless C*-algebra.