This book gives an account of the necessary background for group algebras and crossed products for actions of a group or a semigroup on a space and reports on some very recently developed techniques with applications to particular examples. Much of the material is available here for the first time in book form. The topics discussed are among the most classical and intensely studied C*-algebras. They are important for applications in fields as diverse as the theory of unitary group representations, index theory, the topology of manifolds or ergodic theory of group actions.
Part of the most basic structural information for such a C*-algebra is contained in its K-theory. The determination of the K-groups of C*-algebras constructed from group or semigroup actions is a particularly challenging problem. Paul Baum and Alain Connes proposed a formula for the K-theory of the reduced crossed product for a group action that would permit, in principle, its computation. By work of many hands, the formula has by now been verified for very large classes of groups and this work has led to the development of a host of new techniques. An important ingredient is Kasparov's bivariant K-theory.
More recently, also the C*-algebras generated by the regular representation of a semigroup as well as the crossed products for actions of semigroups by endomorphisms have been studied in more detail.
Intriguing examples of actions of such semigroups come from ergodic theory as well as from algebraic number theory. The computation of the K-theory of the corresponding crossed products needs new techniques. In cases of interest the K-theory of the algebras reflects ergodic theoretic or number theoretic properties of the action.
Joachim Cuntz is a full Professor at the Westfalische Wilhelms-Universitat in Munster, Germany. Siegfried Echterhoff is a full Professor at the Westfalische Wilhelms-Universitat in Munster, Germany. Xin Li is a Senior Lecturer in Pure Mathematics at Queen Mary University of London, United Kingdom. Guoliang Yu is Powell Chair in Mathematics and Professor at the Texas A&M University, USA.
Introduction.- Crossed products and the Mackey-Rieffel-Green machine.- Bivariant KK-Theory and the Baum-Connes conjecure.- Quantitative K-theory for geometric operator algebras.- Semigroup C*-algebras.- Algebraic actions and their C*-algebras.- Semigroup C*-algebras and toric varieties.