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Topological and bivariant K-theory. (English) Zbl 1139.19001
Oberwolfach Seminars 36. Basel: Birkhäuser (ISBN 978-3-7643-8398-5/pbk). xi, 262 p. (2007).
Publisher’s summary: “Topological $$K$$-theory is one of the most important invariants for noncommutative algebras equipped with a suitable topology. Bott periodicity, homotopy invariance, and various long exact sequences distinguish it from algebraic $$K$$-theory. This book describes a bivariant $$K$$-theory for bornological algebras, which provides a vast generalization of topological $$K$$-theory. In addition, it details other approaches to bivariant $$K$$-theories for operator algebras. The book studies a number of applications, including $$K$$-theory of crossed products, the Baum-Connes assembly map, twisted $$K$$-theory with some if its applications, and some variants of the Atiyah-Singer index theorem.”
The book is some review of new trends in the subjects, but it seems to be difficult for beginners: the review of the BDF theory (Chapter 8) and its classical applications to the spectral theory, to the group $$C^*$$-algebras and to the theory of pseudodifferential operators, before the cyclic theories appeared is not enough clarified; Chapter 11 is too short. Maybe the reader could not imagine the importance of the operator $$K$$-theories in the classical applications and in the new application like $$T$$-duality and mirror symmetry in physics.

MSC:
 19-02 Research exposition (monographs, survey articles) pertaining to $$K$$-theory 46L80 $$K$$-theory and operator algebras (including cyclic theory) 46L85 Noncommutative topology 58J20 Index theory and related fixed-point theorems on manifolds 81T75 Noncommutative geometry methods in quantum field theory 19Cxx Steinberg groups and $$K_2$$