The cone of functionals on the Cuntz semigroup

2011/02/07 by Leonel Robert, Robert, Leonel · 4 citations
Mathematics · #FOS: Mathematics #Operator Algebras (math.OA) #math.OA

paper · pdf · doi:10.48550/arxiv.1102.1451

arxiv created 2014/01/06 · arxiv updated 2014/01/07

Abstract

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.

Cited by

• The Global Glimm Property for C*-algebras of topological dimension zero
• Extensions of pure C*-algebras
• The Rank Ratio Function and the Radius of Comparison
• The relative radius of comparison of the crossed product of a non-unital C*-algebra by a finite group

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