C*-Algebras Associated with Iterated Function Systems

TL;DR

This paper reviews the construction of C*-algebras from iterated function systems, establishing connections with Cuntz algebras and Exel's crossed products under certain conditions.

Contribution

It demonstrates how Kajiwara-Watatani algebras relate to Cuntz algebras and Exel's crossed products when specific conditions are met.

Findings

Constructs injective homomorphisms to Cuntz algebras under finite branch or open set conditions.

Describes the image of the homomorphism explicitly.

Shows isomorphism to Exel's crossed product when a left inverse exists.

Abstract

We review Kajiwara and Watatani's construction of a C*-algebra from an iterated function system (IFS). If the IFS satisfies the finite branch condition or the open set condition, we build an injective homomorphism from Kajiwara-Watatani algebras to the Cuntz algebra, which can be thought as the algebra of the lifted system, and we give the description of its image. Finally, if the IFS admits a left inverse we show that the Kajiwara-Watatani algebra is isomorphic to an Exel's crossed product.