C*-Algebras of algebraic dynamical systems and right LCM semigroups

TL;DR

This paper introduces algebraic dynamical systems involving right LCM semigroups acting on groups, constructs associated C*-algebras, and analyzes their properties including functoriality and K-theory, offering new tools for their study.

Contribution

It defines algebraic dynamical systems with semigroup actions, constructs their C*-algebras, and develops the Nica-Toeplitz algebra framework as a novel approach.

Findings

Established functoriality of full semigroup C*-algebras.

Computed K-theory for a broad class of semigroups.

Introduced Nica-Toeplitz algebras as an alternative analysis tool.

Abstract

We introduce algebraic dynamical systems, which consist of an action of a right LCM semigroup by injective endomorphisms of a group. To each algebraic dynamical system we associate a C*-algebra and describe it as a semigroup C*-algebra. As part of our analysis of these C*-algebras we prove results for right LCM semigroups. More precisely we discuss functoriality of the full semigroup C*-algebra and compute its K-theory for a large class of semigroups. We introduce the notion of a Nica-Toeplitz algebra of a product system over a right LCM semigroup, and show that it provides a useful alternative to study algebraic dynamical systems.