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

Abstract

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 $K_1(C)=\{0\}.$ Suppose that $\lambda :{\mathrm{Cu}}^{\sim }(C)\to {\mathrm{Cu}}^{\sim }(A)$ is a morphism in Cuntz semigroups which is strictly positive. Then there exists a sequence of homomorphisms ${\varphi }_n:C\to A$ such that ${\lim }_{n\to \infty }{\mathrm{Cu}}^{\sim }({\varphi }_n)=\lambda .$ 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.

A few correction were made

Cited by in corpus (2)

• On classification of non-unital amenable simple C*-algebras, III, the range and the reduction
• Equivariant Kirchberg-Phillips type absorption for the Razak-Jacelon algebra