Purely infinite C*-Algebras: Ideal-preserving zero homotopies

Abstract.

We show that if A is a separable, nuclear, $\mathcal{O}_\infty$-absorbing (or strongly purely infinite) C*-algebra which is homotopic to zero in an ideal-system preserving way, then A is the inductive limit of C*-algebras of the form $C_0 (\Gamma ,\upsilon ) \otimes M_k ,$ where Γ is a finite connected graph (and $C_0 (\Gamma ,\upsilon )$ is the algebra of continuous functions on Γ that vanish at a distinguished point $\upsilon \in \Gamma$).

We show further that if B is any separable, nuclear C*-algebra, then $B \otimes \mathcal{O}_2 \otimes \mathcal{K}$ is isomorphic to a crossed product $D \rtimes_{\alpha} \mathbb{Z},$ where D is an inductive limit of C*-algebras of the form $C_0 (\Gamma ,\upsilon ) \otimes M_k$ (and D is $\mathcal{O}_2$ -absorbing and homotopic to zero in an ideal-system preserving way).