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.
