Computable presentations of C*-algebras

TL;DR

This paper explores the computable presentations of C*-algebras within effective metric structure theory, establishing foundational notions and contrasting their properties with those of groups, including categoricity results.

Contribution

It introduces notions of recursive presentations and word problems for C*-algebras and demonstrates that finite-dimensional C*-algebras are computably categorical, unlike some groups.

Findings

Finite-dimensional C*-algebras are computably categorical.

Not all C*-algebras share the computable categoricity of finitely generated groups.

Analogous results to group theory are established for C*-algebras.

Abstract

We initiate the study of computable presentations of real and complex C*-algebras under the program of effective metric structure theory. With the group situation as a model, we develop corresponding notions of recursive presentations and word problems for C*-algebras, and show some analogous results hold in this setting. Famously, every finitely generated group with a computable presentation is computably categorical, but we provide a counterexample in the case of C*-algebras. On the other hand, we show every finite-dimensional C*-algebra is computably categorical.