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

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.

25 pages

References in corpus (4)

• KMS states on the C*-algebras of non-principal groupoids
• The Regular C*-algebra of an Integral Domain
• C*-algebras associated with endomorphisms of groups
• $C^{\ast }$-algebras associated with the $ax+b$-semigroup over $\mathbb{N}$

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• On the K-theory of crossed products by automorphic semigroup actions
• Almost mixing of all orders and CLT for some ${\mathbb{Z}}^d$-actions on subgroups of $\mathbb{F}\_p^{{\mathbb{Z}}^d}$
• Algebraic actions I. C*-algebras and groupoids
• Interactions and dynamical systems of type (n,m) - A case study
• Universal C*-algebras of endomorphisms of groups and partial actions
• The bicategory of groupoid correspondences
• On Cohomology for Product Systems