The cone of functionals on the Cuntz semigroup

Details The cone of functionals on the Cuntz semigroup The cone of functionals on the Cuntz semigroup

2011-02-07

arxiv.org/abs/1102.1451v3

Mathematics

Operator Algebras

Added to the hypotheses of several results that the semigroup has the property of almost algebraic order (and made the necessa

Scientific paper

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.

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