On the C*-envelope of approximately finite-dimensional operator algebras

TL;DR

This paper proves that the C*-envelope of certain limit algebras derived from digraph systems is an approximately finite-dimensional C*-algebra and explicitly describes its direct system.

Contribution

It establishes that the C*-envelope of limit algebras from contractive regular digraph systems is AF and provides an explicit construction of its direct system.

Findings

C*-envelope is an AF C*-algebra

Explicit description of the direct system for the C*-envelope

Extension of AF algebra theory to digraph algebra limits

Abstract

The C*-envelope of the limit algebra (or limit space) of a contractive regular system of digraph algebras (or digraph spaces) is shown to be an approximately finite C*-algebra and the direct system for the C*-envelope is determined explicitly.