C*-Algebras Associated with Endomorphisms and Polymorphisms of Compact Abelian Groups

A surjective endomorphism or, more generally, a polymorphism in the sense of Schmidt and Vershik [Erg Th Dyn Sys 28(2):633–642, 2008], of a compact abelian group H induces a transformation of L2(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 L2(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.