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

Christopher J. Eagle, Isaac Goldbring, Timothy H. McNicholl

TL;DR

This work develops a comprehensive framework for computable K-theory of AF algebras, showing that every computably presented AF algebra admits a computable AF certificate and that the computable $K_0$-functor yields a computable equivalence between c.e. presentations of unital AF algebras and unital dimension groups. Using an effective version of Glimm'sLemma and Shen-type properties, the authors prove effective versions of the Effros-Handelman-Shen theorem and Elliott classification, enabling computable reconstruction of algebras from their dimension groups and Bratteli diagrams. The paper then establishes a computable equivalence of categories between the two computable presentation frameworks and derives complexity results for index sets and isomorphism problems in AF and UHF algebras. Overall, the results provide uniform procedures for moving between algebraic, combinatorial, and categorical descriptions in the computable setting, with broad implications for effective classification in operator algebras. The findings significantly advance the interface of computability theory with $C^*$-algebra classification, offering concrete algorithms and complexity bounds for effective presentations and isomorphisms.

Abstract

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.

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

TL;DR

This work develops a comprehensive framework for computable K-theory of AF algebras, showing that every computably presented AF algebra admits a computable AF certificate and that the computable -functor yields a computable equivalence between c.e. presentations of unital AF algebras and unital dimension groups. Using an effective version of Glimm'sLemma and Shen-type properties, the authors prove effective versions of the Effros-Handelman-Shen theorem and Elliott classification, enabling computable reconstruction of algebras from their dimension groups and Bratteli diagrams. The paper then establishes a computable equivalence of categories between the two computable presentation frameworks and derives complexity results for index sets and isomorphism problems in AF and UHF algebras. Overall, the results provide uniform procedures for moving between algebraic, combinatorial, and categorical descriptions in the computable setting, with broad implications for effective classification in operator algebras. The findings significantly advance the interface of computability theory with -algebra classification, offering concrete algorithms and complexity bounds for effective presentations and isomorphisms.

Abstract

We continue the study of the effective content of -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 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.
Paper Structure (59 sections, 52 theorems, 34 equations)

This paper contains 59 sections, 52 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Background from classical mathematics
  3. Inductive limits in categories
  4. Ordered abelian groups
  5. Basic definitions and principles
  6. Simplicial groups
  7. Inductive limits
  8. Dimension groups
  9. $\mathrm{C}^*$-algebras
  10. General background
  11. $K$-theory
  12. Matrix algebras
  13. Finite-dimensional $\mathrm{C}^*$-algebras
  14. AF algebras
  15. Some perturbation principles
  16. ...and 44 more sections

Key Result

Proposition 2.1

Suppose $(A, (\nu_n)_{n \in \mathbb{N}})$ and $(B, (\mu_n)_{n \in \mathbb{N}})$ are inductive limits of $(A_n, \phi_n)_{n \in \mathbb{N}}$. Then the reduction from $(A, (\nu_n)_{n \in \mathbb{N}})$ to $(B, (\mu_n)_{n \in \mathbb{N}})$ is an isomorphism.

Theorems & Definitions (115)

  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • Theorem 2.4: Classification of finite-dimensional algebras
  • Definition 2.5
  • Theorem 2.6: Elliott's Theorem
  • Lemma 2.7
  • Lemma 2.8
  • proof
  • ...and 105 more