The classification problem for amenable C-algebras · 1994-ana-174

Source-reported assessment from the ICM catalogue snapshot. This is not an independent review by Frontier Science.

The problem is unresolved. No proof or counterexample is known under exactly the stated hypotheses.

The strongest complete positive result requires extra hypotheses. If $A$ is also purely infinite and satisfies the Universal Coefficient Theorem (UCT), the Kirchberg-Phillips classification theorem identifies unital Kirchberg algebras by their graded $K$-groups and the class of the unit. Thus $K_0(A)=K_1(A)=0$ implies $A\cong\mathcal O_2$. Mikael Rørdam further proved that a separable nuclear $\mathcal Z$-stable algebra is $\mathcal O_\infty$-stable exactly when it is traceless. Consequently, the assertion holds when both UCT and $\mathcal Z$-stability are added.

Even the narrower case in which $A$ is assumed purely infinite remains open without UCT. Rørdam and Størmer, Corollary 8.4.6(ii), show that the assertion that every unital Kirchberg algebra with $K_*(A)=0$ is isomorphic to $\mathcal O_2$ is equivalent to the UCT problem for nuclear $C^*$-algebras. Schafhauser, Tikuisis, and White explicitly record this equivalence and the continuing open status in their current survey. Hence the original statement, which does not assume UCT, contains a major unresolved special case.

There is also no general route from the original no-trace hypothesis to pure infiniteness. Rørdam constructed a simple, separable, nuclear UCT algebra containing both a nonzero finite projection and an infinite projection; it has no tracial state but is not purely infinite. This disproves the general implication “traceless implies purely infinite,” although that construction does not establish a counterexample having both $K$-groups zero. Thus it does not resolve the exact problem.

Key references:

• George A. Elliott, “The Classification Problem for Amenable $C^*$-Algebras,” Proceedings of the International Congress of Mathematicians, Zürich 1994, Birkhäuser, 1995, pp. 922-932, DOI: 10.1007/978-3-0348-9078-6_85. Section 6 states the exact proposed characterization of $\mathcal O_2$.
• N. Christopher Phillips, “A Classification Theorem for Nuclear Purely Infinite Simple $C^*$-Algebras,” Documenta Mathematica 5 (2000), 49-114, arXiv:funct-an/9506010. It proves the UCT purely infinite case via the Kirchberg-Phillips classification theorem.
• Mikael Rørdam and Erling Størmer, Classification of Nuclear $C^*$-Algebras. Entropy in Operator Algebras, Encyclopaedia of Mathematical Sciences 126, Springer, 2002, DOI: 10.1007/978-3-662-04825-2. Corollary 8.4.6(ii) gives the equivalence between the UCT problem and uniqueness of $\mathcal O_2$ among unital Kirchberg algebras with zero $K$-theory.
• Mikael Rørdam, “A Simple $C^*$-Algebra with a Finite and an Infinite Projection,” Acta Mathematica 191 (2003), 109-142, DOI: 10.1007/BF02392697, arXiv:math/0204339. Theorem 6.10 and Corollary 7.1 give a separable nuclear UCT example with both finite and infinite projections, showing that absence of traces alone does not force pure infiniteness.
• Mikael Rørdam, “The Stable and the Real Rank of $\mathcal Z$-Absorbing $C^*$-Algebras,” International Journal of Mathematics 15 (2004), 1065-1084, arXiv:math/0408020. Theorem 5.2 proves that separable nuclear $\mathcal Z$-stable traceless algebras are $\mathcal O_\infty$-stable, giving the principal regularity-based partial result.
• Christopher Schafhauser, Aaron Tikuisis, and Stuart White, “Nuclear $C^*$-Algebras: 99 Problems,” preprint, 2025 (revised 2026), arXiv:2506.10902. Section 3 records that the zero-$K$-theory Kirchberg special case is equivalent to the still-open UCT problem.