A new proof of Kirchberg's $\mathcal O_2$-stable classification

TL;DR

This paper offers a new proof of Kirchberg's classification theorem for certain $C^*$-algebras, showing they are isomorphic if their ideal structures are equivalent, with many results not relying on pure infiniteness.

Contribution

The paper provides a novel proof of Kirchberg's $ ext{O}_2$-stable classification theorem, broadening understanding without depending on pure infiniteness assumptions.

Findings

Classification based on ideal lattice isomorphism

Many intermediate results do not depend on pure infiniteness

Establishes equivalence of isomorphism and primitive ideal space homeomorphism

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.