Algebraic actions I. C*-algebras and groupoids

TL;DR

This paper develops a framework linking algebraic actions of semigroups to associated C*-algebras and groupoids, providing conditions for properties like Hausdorffness and topological freeness, and analyzing their structural features.

Contribution

It introduces a novel approach to study C*-algebras from algebraic semigroup actions via inverse semigroups and groupoids, characterizing key properties and their implications.

Findings

Groupoids are purely infinite when minimal.

Topologically free groupoids yield C*-algebras between full and essential versions.

Structural results for C*-algebras from shifts, algebraic actions, and rings.

Abstract

We provide a framework for studying concrete C*-algebras associated with algebraic actions of semigroups: Given such an action, we construct an inverse semigroup, and we introduce conditions for algebraic actions that characterize Hausdorffness, topological freeness, and minimality of the associated tight groupoid. We parameterize all closed invariant subspaces of the unit space of our groupoid, and characterize topological freeness of the associated reduction groupoids. We prove that our groupoids are purely infinite whenever they are minimal, and in the topologically free case, we prove that our concrete C*-algebra is always a (possibly exotic) groupoid C*-algebra in the sense that it sits between the full and essential C*-algebras of our groupoid. As an application, we obtain structural results for C*-algebras associated with, for instance, shifts over semigroups, actions coming from commutative algebra, and non-commutative rings.