Traces on cores of C*-algebras associated with self-similar maps

TL;DR

This paper classifies all extreme tracial states on the cores of C*-algebras linked to self-similar maps, identifying both discrete and continuous types explicitly.

Contribution

It provides a complete classification and explicit list of extreme tracial states for these C*-algebras, combining discrete and continuous measures.

Findings

Extreme tracial states are classified into discrete and continuous types.

Discrete states are supported on finite orbits of branch points.

Continuous states are given by the Hutchinson measure on the self-similar set.

Abstract

We completely classify the extreme tracial states onthe cores of the C*-algebras associated with self-similar maps on compact metric spaces. We present a complete list of them. The extreme tracial states are the union of the discrete type tracial states given by measures supported on the finite orbits of the branch points and a continuous type tracial state given by the Hutchinson measure on the original self-similar set.