C*-algebras associated with endomorphisms and polymorphisms of compact abelian groups

TL;DR

This paper investigates C*-algebras generated by transformations induced by endomorphisms and polymorphisms of compact abelian groups, providing conditions for simplicity, pure infiniteness, and computing their K-theory.

Contribution

It introduces a framework for describing these C*-algebras via generators and relations, and computes their K-theory for various examples.

Findings

The algebras are simple under natural conditions.

Endomorphism-based algebras are always purely infinite.

K-theory formulas enable explicit computations in examples.

Abstract

A surjective endomorphism or, more generally, a polymorphism in the sense of \cite{SV}, of a compact abelian group $H$ induces a transformation of $L^2(H)$. We study the C*-algebra generated by this operator together with the algebra of continuous functions $C(H)$ which acts as multiplication operators on $L^2(H)$. Under a natural condition on the endo- or polymorphism, this algebra is simple and can be described by generators and relations. In the case of an endomorphism it is always purely infinite, while for a polymorphism in the class we consider, it is either purely infinite or has a unique trace. We prove a formula allowing to determine the $K$-theory of these algebras and use it to compute the $K$-groups in a number of interesting examples.