A topological isomorphism invariant for certain AF algebras

Ryan J Zerr

doi:10.1155/ijmms.2005.1665

Abstract

For certain AF algebras, a topological space is described which provides an isomorphism invariant for the algebras in this class. These AF algebras can be described in graphical terms by virtue of the existence of a certain type of Bratteli diagram, and the order‐preserving automorphisms of the corresponding AF algebra′s dimension group are then studied by utilizing this graph. This will also provide information about the automorphism groups of the corresponding AF algebras.

Highlights

In studying approximately finite-dimensional (AF) C∗-algebras, [2] introduced an infinite graph, called a Bratteli diagram, which can be used to describe the structure of the algebra
These graphs have come to play an important role in the theory of AF algebras, and in this paper, we will be concerned with a certain class of Bratteli diagrams which have natural subgraphs that provide isomorphism invariants for the corresponding AF algebras
In addition to the information it will provide about AF algebras, the invariant described here will be useful in that it will allow us to say something about the orderpreserving automorphisms on these dimension groups

Summary

A TOPOLOGICAL ISOMORPHISM INVARIANT FOR CERTAIN AF ALGEBRAS

For certain AF algebras, a topological space is described which provides an isomorphism invariant for the algebras in this class. These AF algebras can be described in graphical terms by virtue of the existence of a certain type of Bratteli diagram, and the orderpreserving automorphisms of the corresponding AF algebra’s dimension group are studied by utilizing this graph. This will provide information about the automorphism groups of the corresponding AF algebras

Introduction

Dimension groups of continuous functions

Applications and further questions