On C*-algebras of irreversible algebraic dynamical systems

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 turns out to be closely related to generalized 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.