A new proof of Kirchberg's ${\mathcal{O}}_2$-stable classification

Abstract

I present a new proof of Kirchberg's ${\mathcal{O}}_2$-stable classification theorem: two separable, nuclear, stable/unital, ${\mathcal{O}}_2$-stable $C^{\ast }$-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.

Final version, to appear in J. Reine Angew. Math., 39 pages

References in corpus (2)

• Completely positive maps of order zero
• Classification of finite simple amenable $\mathcal{Z}$-stable $C^{\ast }$-algebras

Cited by in corpus (4)

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• The nuclear dimension of ${\mathcal{O}}_{\infty }$-stable $C^{\ast }$-algebras
• Decomposing nuclear maps
• Rohlin actions of finite groups on the Razak-Jacelon algebra