The Cone of Functionals on the Cuntz Semigroup

Abstract

The functionals on an ordered semigroup $S$ in the category $\mathbf{Cu}$ - a category to which the Cuntz semigroup of a C*-algebra naturally belongs - are investigated. After appending a new axiom to the category $\mathbf{Cu}$, it is shown that the "realification" $S_{\mathsf{R}}$ 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 $S_{\mathsf{R}}$ 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.