Weak pure infiniteness and Cuntz-algebra absorption

Resolves the ordinary-to-strong pure-infiniteness question of Kirchberg and Rørdam for complex C^∗-algebras. For exact algebras, proper infiniteness of one fixed finite amplification of every positive element also suffices. Consequently, every separable nuclear algebra with this property absorbs \mathcal O_\infty, without unitality or simplicity assumptions. (Lean)

Manuscript

Weak pure infiniteness and O-infinity absorption

September 25, 2026 · source · cite as \cite{OAI:Weak-pure-infiniteness-and-O-infinity-absorption-September-25-2026}

We prove that every complex C^∗-algebra in which each positive element is properly infinite is strongly purely infinite. This answers the ordinary-to-strong part of Kirchberg–Rørdam's comparison question. For exact algebras, we also prove that proper infiniteness of one fixed finite amplification of every positive element implies proper infiniteness of each positive element. Consequently, every separable nuclear algebra with this fixed-amplification property absorbs \mathcal O_\infty, without assumptions of unitality or simplicity.

Result 303 · Operator algebras. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.