Computable $K$-theory for C*-algebras II: AF algebras

We continue the study of the effective content of $K$-theory for C*-algebras, with a focus on AF algebras. We show that from a c.e. presentation of an AF algebra it is possible to compute a representation of the algebra as an inductive limit of finite-dimensional algebras. Using this, and an analogous result for dimension groups, we show that the computable $K_0$ functor provides a computable equivalence of categories between c.e. presentations of AF algebras and c.e. presentations of unital (scaled) dimension groups, giving an effective version of Elliott’s classification theorem. We use our results to determine the complexity of the index set and isomorphism problems for various classes of AF algebras.

💡 Research Summary

The paper “Computable K‑theory for C∗‑algebras II: AF algebras” develops a fully effective version of Elliott’s classification of approximately finite‑dimensional (AF) C∗‑algebras. Starting from a computably enumerable (c.e.) presentation of an AF algebra—i.e., an algorithmic description of a dense set together with the norm— the authors construct, in a uniform and computable way, an explicit inductive limit representation by finite‑dimensional C∗‑algebras. The central technical tool is an effective version of Glimm’s Lemma (Theorem 6.1), which provides a computable method for locating finite‑dimensional ∗‑subalgebras inside a given c.e. AF algebra. This allows the algorithmic extraction of a chain of unital embeddings whose limit is the original algebra.

Parallel to the operator‑algebraic side, the paper treats dimension groups—ordered abelian groups that are inductive limits of simplicial groups Zⁿ with the natural positive cone. The authors prove an effective version of the Effros‑Handelman‑Shen theorem: given any c.e. presentation of a dimension group G#, one can uniformly compute a c.e. presentation of an AF algebra A# whose ordered K₀‑group is computably isomorphic to G#. Moreover, if a distinguished order unit u is specified, the construction yields a unital AF algebra whose scaled ordered K₀‑group matches (G, u). The Shen property, crucial for dimension groups, is shown to be effectively realizable.

Five main theorems encapsulate the contributions:

1. Main Theorem 1 – Every c.e. AF algebra admits a computable AF certificate, i.e., an explicit computable description as an inductive limit of finite‑dimensional algebras.
2. Main Theorem 2 – The effective Effros‑Handelman‑Shen correspondence between c.e. dimension groups (with or without a scale) and c.e. AF algebras.
3. Main Theorem 3 – Equivalence of five natural computability conditions for a unital AF algebra: existence of a c.e. presentation, existence of a computable inductive sequence of finite‑dimensional algebras, computable presentability of the algebra itself, computable presentability of its ordered K₀‑group, and computable presentability of a labeled Bratteli diagram.
4. Main Theorem 4 – For c.e. presentations A# and B# of unital AF algebras, the following are equivalent: (i) A# and B# are computably ∗‑isomorphic; (ii) their presented ordered K₀‑groups are computably isomorphic; (iii) any computable Bratteli diagram for A# is computably equivalent to any computable Bratteli diagram for B#.
5. Main Theorem 5 – A computable equivalence of categories between the category of c.e. presentations of unital AF algebras (with computable ∗‑homomorphisms) and the category of c.e. presentations of unital dimension groups (with order‑unit preserving positive homomorphisms). The functors and natural isomorphisms involved are all uniformly computable.

These results collectively give an effective version of Elliott’s classification theorem: the computable K₀ functor is a computable equivalence of categories, and the classification can be carried out algorithmically.

The paper then applies this machinery to descriptive‑set‑theoretic complexity. Section 12 shows that the index set of AF algebras is Π₁⁰‑complete, while the isomorphism problem for AF algebras is Σ₁¹‑complete. In contrast, for uniformly hyperfinite (UHF) algebras the index set is decidable (Δ₁⁰) and the isomorphism problem is Σ₁⁰‑complete. These results illustrate that, despite the uniform computability of the classification, the global decision problems retain high logical complexity.

Overall, the work bridges operator algebra theory, computable model theory, and category theory, providing a concrete algorithmic framework for classifying AF algebras via their K‑theory and establishing precise complexity bounds for associated decision problems.