Unitary groups and augmented Cuntz semigroups of separable simple Z-stable C*-algebras

Abstract

Let $A$ be a separable simple exact ${\cal Z}$-stable $C^*$-algebra. We show that the unitay group of ${\tilde A}$ has the cancellation property. If $A$ has continuous scale, the Cuntz semigroup of $\tilde A$ has the strict comparison property and a weak cancellation property. Let $C$ be a 1-dimensional non-commutative CW complex with $K_1(C)=\{0\}.$ Suppose that $\lambda: {\rm Cu}^\sim(C)\to {\rm Cu}^\sim(A)$ is a morphism in Cuntz semigroups which is strictly positive. Then there exists a sequence of homomorphisms $\phi_n: C\to A$ such that $\lim_{n\to\infty}{\rm Cu}^\sim(\phi_n)=\lambda.$ This result leads to the proof that every separable amenable simple $C^*$-algebra in the UCT class has rationally generalized tracial rank at most one.