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K-Theory for Group C*-Algebras and Semigroup C*-Algebras (Paperback, 1st ed. 2017)
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K-Theory for Group C*-Algebras and Semigroup C*-Algebras (Paperback, 1st ed. 2017)
Series: Oberwolfach Seminars, 47
Expected to ship within 10 - 15 working days
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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.
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