The classification of some generalised Bunce-Deddens algebras

TL;DR

This paper uses K-theory to classify a broad class of generalized Bunce-Deddens algebras, demonstrating that supernatural numbers serve as complete invariants for their isomorphism classes.

Contribution

It provides an isomorphism theorem for generalized Bunce-Deddens algebras and establishes supernatural numbers as complete invariants for classification.

Findings

Computed the torsion-free part of K_0 for these algebras.

Proved supernatural numbers classify these algebras up to isomorphism.

Extended classification results to a large class of algebras.

Abstract

We use $K$-theory to prove an isomorphism theorem for a large class of generalised Bunce-Deddens algebras constructed by Kribs and Solel from a directed graph $E$ and a sequence $\omega$ of positive integers. In particular, we compute the torsion-free component of the $K_0$-group for a class of generalised Bunce-Deddens algebras to show that supernatural numbers are a complete invariant for this class.