Abstract
AbstractWe prove that Cuntz semigroups of C*-algebras satisfy Edwards' condition with respect to every quasitrace. This condition is a key ingredient in the study of the realization problem of functions on the cone of quasitraces as ranks of positive elements. In the course of our investigation, we identify additional structure of the Cuntz semigroup of an arbitrary C*-algebra and of the cone of quasitraces.
Highlights
The rank of a positive element a in a C∗-algebra A with respect to a trace τ is defined as dτ (a) = limn τ (a1/n)
In case of a trace, this rank is nothing but the value of the support projection of a in A∗∗ under the canonical extension of τ to a normal trace on A∗∗; see [18]
Property (O7) We identify a new property that Cuntz semigroups of C∗-algebras satisfy
Summary
Given an extreme quasitrace τ in QT1(A), it was shown in [23, theorem 4.7] that for any two positive elements a and b in A, the minimum of the ranks of a and b with respect to τ can be approximated by the ranks of positive elements c that are dominated by a and b in the sense of Cuntz: min dτ (a), dτ (b) = sup dτ (c) : c a, b This property was termed Edwards’ condition for τ by the fourth named author due to its relation with the work in [11]. We prove that the cone of quasitraces on a C∗-algebra A satisfies Riesz refinement, proposition 3.3, a result which is significant towards establishing Edwards’ condition for Cuntz semigroups of C∗-algebras. By the fourth author in [23] combined with the results obtained in the previous sections
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