Certain Cuntz semigroup properties of extension C*-algebras

Abstract

Let $0\to I\to A\to A/I\to 0$ be a short exact sequence of C*-algebras. In this paper, we prove that if $\dim (\mathrm{Cu}(I))\le n$ and $\dim (\mathrm{Cu}(A/I))\le m$, then $\dim (\mathrm{Cu}(A))\le n+m+1$, this result gives an affirmative answer to a problem concerning the Cu-semigroup of C*-algebras raised by Thiel and Vilalta. We further show that certain comparison and divisibility properties satisfied by $I$ and $A/I$ are inherited by the extension algebra $A$. As an application, we provide an alternative proof of equivalence that $A$ is pure if and only if $I$ and $A/I$, a result originally proven by Perera, Thiel, and Vilalta.

arXiv comment18 pages