Ideals of the core of C*-algebras associated with self-similar maps

TL;DR

This paper classifies the ideals of the core of C*-algebras linked to self-similar maps, showing they are determined by the map's singularity structure and identifying conditions for simplicity.

Contribution

It provides a complete classification of these ideals and relates their structure to the singularities of the self-similar map, using matrix representations.

Findings

Ideals are determined by intersections with the coefficient algebra C(K).

The core is simple iff the self-similar map has no branch points.

The classification is achieved via matrix representations of the core.

Abstract

We give a complete classification of the ideals of the core of the C*-algebras associated with self-similar maps under a certain condition. Any ideal is completely determined by the intersection with the coefficient algebra C(K) of the self-similar set K. The corresponding closed subset of K is described by the singularity structure of the self-similar map. In particular the core is simple if and only if the self-similar map has no branch point. A matrix representation of the core is essentially used to prove the classification.