Kirchberg-Rordam comparison question: pure infiniteness implies strong pure infiniteness, and weak equals strong for exact C*-algebras

Theorem 1.1: every complex C*-algebra in which every positive element is properly infinite is strongly purely infinite, with an arbitrary cross term in the diagonalization. Theorem 1.2: for exact algebras, proper infiniteness of $h^{\oplus n}$ for one fixed $n$ and all positive $h$ implies proper infiniteness of every positive element, hence strong pure infiniteness. Corollary: a separable nuclear algebra with this fixed-amplification property absorbs $\mathcal O_\infty$, with no unitality or simplicity assumption (via Kirchberg's central-sequence results). Not settled: whether weak pure infiniteness implies pure infiniteness for non-exact algebras, which the paper states remains outside its conclusions.

Theorem 1.1: every complex C*-algebra in which every positive element is properly infinite is strongly purely infinite, with an arbitrary cross term in the diagonalization. Theorem 1.2: for exact algebras, proper infiniteness of $h^{\oplus n}$ for one fixed $n$ and all positive $h$ implies proper infiniteness of every positive element, hence strong pure infiniteness. Corollary: a separable nuclear algebra with this fixed-amplification property absorbs $\mathcal O_\infty$, with no unitality or simplicity assumption (via Kirchberg's central-sequence results). Not settled: whether weak pure infiniteness implies pure infiniteness for non-exact algebras, which the paper states remains outside its conclusions.