On C*-algebras of irreversible algebraic dynamical systems

TL;DR

This paper introduces a framework for irreversible algebraic dynamical systems, associating universal C*-algebras that are Kirchberg algebras, and explores their structure and related product systems.

Contribution

It develops a general notion of independence for group endomorphisms and constructs associated C*-algebras as Cuntz-Nica-Pimsner algebras, extending prior work on dynamical systems.

Findings

Universal C*-algebras are UCT Kirchberg algebras under natural conditions.

The structure of the core subalgebra relates to generalized Bunce-Deddens algebras.

A decomposition theorem for semigroup crossed products is established.

Abstract

Extending the work of Cuntz and Vershik, we develop a general notion of independence for commuting group endomorphisms. Based on this concept, we initiate the study of irreversible algebraic dynamical systems, which can be thought of as irreversible analogues of the dynamical systems considered by Schmidt. To each irreversible algebraic dynamical system, we associate a universal C*-algebra and show that it is a UCT Kirchberg algebra under natural assumptions. Moreover, we discuss the structure of the core subalgebra, which is closely related to generalised Bunce-Deddens algebras in the sense of Orfanos. We also construct discrete product systems of Hilbert bimodules for irreversible algebraic dynamical systems which allow us to view the associated C*-algebras as Cuntz-Nica-Pimsner algebras. Besides, we prove a decomposition theorem for semigroup crossed products of unital C*-algebras by semidirect products of discrete, left cancellative monoids.