Perforation conditions and almost algebraic order in Cuntz semigroups

2016/04/22 by Antoine, Ramon, Perera, Francesc, Petzka, Henning
#06B35 #15A69 #46L05 #46L35 #46L80 #FOS: Mathematics #K-Theory and Homology (math.KT) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1604.06718

Abstract

For a C^*-algebra A, it is an important problem to determine the Cuntz semigroup Cu(A\otimesZ) in terms of Cu(A). We approach this problem from the point of view of semigroup tensor products in the category of abstract Cuntz semigroups, by analysing the passage of significant properties from Cu(A) to Cu(A)⊗_CuCu(Z). We describe the effect of the natural map Cu(A)\toCu(A)⊗_CuCu(Z) in the order of Cu(A), and show that, if A has real rank zero and no elementary subquotients, Cu(A)⊗_CuCu(Z) enjoys the corresponding property of having a dense set of (equivalence classes of) projections. In the simple, nonelementary, real rank zero and stable rank one situation, our investigations lead us to identify almost unperforation for projections with the fact that tensoring with Z is inert at the level of the Cuntz semigroup.

Related

• The cone of functionals on the Cuntz semigroup
• The modern theory of Cuntz semigroups of C*-algebras
• The Cuntz semigroup of the tensor product C*-algebras
• A total Cuntz semigroup for C^*-algebras of stable rank one
• Unitary groups and augmented Cuntz semigroups of separable simple Z-stable C*-algebras