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

Let $A$ be a separable simple exact $\mathcal{Z}$-stable $C^{\ast }$-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 $K_1(C)=\{0\}\mathrm{.}$ Suppose that $\lambda: {\rm Cu}^\sim©\to {\rm Cu}^\sim(A)$ is a morphism in Cuntz semigroups which is strictly positive. Then there exists a sequence of homomorphisms $\phi_n: C\to A$suchthat$\lim_{n\to\infty}{\rm Cu}^\sim(\phi_n)=\lambda.$ This result leads to the proof that every separable amenable simple $C^{\ast }$-algebra in the UCT class has rationally generalized tracial rank at most one.