Unitary groups and augmented Cuntz semigroups of separable simple 𝒵-Stable C∗-algebras

Abstract

Let [Formula: see text] be a separable simple exact [Formula: see text]-stable [Formula: see text]-algebra. We show that the unitary group of [Formula: see text] has the cancellation property. If [Formula: see text] has continuous scale then the Cuntz semigroup of [Formula: see text] has strict comparison property and a weak cancellation property. Let [Formula: see text] be a 1-dimensional noncommutative CW complex with [Formula: see text] Suppose that [Formula: see text] is a morphism in the augmented Cuntz semigroups which is strictly positive. Then there exists a sequence of homomorphisms [Formula: see text] such that [Formula: see text] This result leads to the proof that every separable amenable simple [Formula: see text]-algebra[Formula: see text]in the UCT class has rationally generalized tracial rank at most one.