Remark: Family 303. From ordinary to strong pure infiniteness

Resolves the ordinary-to-strong pure-infiniteness question of Kirchberg and Rørdam for complex $C^{\ast }$-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.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.