I present a new proof of Kirchberg’s -stable classification theorem: two separable, nuclear, stable/unital, -stable -algebras are isomorphic if and only if their ideal lattices are order isomorphic, or equivalently, their primitive ideal spaces are homeomorphic. Many intermediate results do not depend on pure infiniteness of any sort.
Funding statement: This work was partially funded by the Carlsberg Foundation through an Internationalisation Fellowship.
Parts of this paper were completed during a research visit at the Mittag–Leffler Institute during the programme Classification of Operator Algebras: Complexity, Rigidity, and Dynamics, and during a visit at the CRM institute during the programme IRP Operator Algebras: Dynamics and Interactions. I am thankful for their hospitality during these visits. I am very grateful to Joan Bosa, Jorge Castillejos, Aidan Sims and Stuart White for valuable, inspiring conversations on topics of the paper. I would also like to thank the referee for many helpful comments and suggestions.
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