Weak pure infiniteness and Cuntz-algebra absorption

What OpenAI claims

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 𝒪_∞, without unitality or simplicity assumptions. (Lean)

Paper

1. Weak pure infiniteness and O-infinity absorption

   25 September 2026 · Preprint, not peer reviewed · PDF

   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 𝒪_∞, without assumptions of unitality or simplicity.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.