Certain Cuntz semigroup properties of extension C*-algebras

Let $0\stackrel{}{\to }I\stackrel{}{\to }A\stackrel{}{\to }A\mathrm{/}I\stackrel{}{\to }0$ be a short exact sequence of C*-algebras. In this paper, we prove that if $\dim (\mathrm{C}\mathrm{u}(I))\le n$ and $\dim (\mathrm{C}\mathrm{u}(A\mathrm{/}I))\le m$, then $\dim (\mathrm{C}\mathrm{u}(A))\le n+m+1$, 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 $A\mathrm{/}I$ 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 $A\mathrm{/}I$, a result originally proven by Perera, Thiel, and Vilalta.