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.

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.

Reviewer: Do Ngoc Diep (Hanoi)

##### 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 \(K_2\) |