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

TL;DR

This paper develops a computable framework for classifying AF algebras via $K$-theory, enabling effective computation of algebra representations and classification problems.

Contribution

It provides a computable equivalence between presentations of AF algebras and dimension groups, extending Elliott's classification theorem effectively.

Findings

Computable $K_0$ functor yields an effective classification of AF algebras.

Determines the complexity of index set and isomorphism problems for AF algebras.

Shows that from a c.e. presentation, an AF algebra can be effectively represented as an inductive limit.

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.