C*-algebras associated with algebraic actions

TL;DR

This survey reviews the construction and analysis of C*-algebras derived from algebraic endomorphisms of abelian groups, emphasizing the action of Dedekind ring semigroups and their associated mathematical challenges.

Contribution

It highlights the role of algebraic actions of Dedekind semigroup on abelian groups in shaping new methods for C*-algebra analysis.

Findings

Focus on algebraic actions of Dedekind semigroups

Development of new methods for C*-algebra analysis

Identification of intriguing problems in algebraic representations

Abstract

This is a survey of work in which the author was involved in recent years. We consider C*-algebras constructed from representations of one or several algebraic endomorphisms of a compact abelian group - or, dually, of a discrete abelian group. In our survey we do not try to describe the entire scope of the methods and results obtained in the original papers, but we concentrate on the important thread coming from the action of the multiplicative semigroup of a Dedekind ring on its additive group. Representations of such actions give rise to particularly intriguing problems and the study of the corresponding C*-algebras has motivated many of the new methods and general results obtained in this area.