Crossed products of \(C\)*-algebras.

*(English)*Zbl 1119.46002
Mathematical Surveys and Monographs 134. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-4242-0/hbk). xvi, 528 p. (2007).

Crossed products trace their origins back to statistical mechanics and to the group measure space constructions of Murray and von Neumann. Crossed products of \(C^*\)-algebras are both a source of new examples of \(C^*\)-algebras and are interesting in their own right. The aim of this nicely written and self-contained book is to provide the tools necessary to begin doing research involving crossed product \(C^*\)-algebras.

The book begins with an overview of the group theory of locally compact groups and Haar measure. Then dynamical systems and their associated crossed products are defined (Chapter 2), the structure of the group \(C^*\)-algebras of Abelian and compact groups is worked out and some basic tools are developed (Chapter 3). One of the main results of Chapter 4 is Raeburn’s symmetric imprimitivity theorem, which provides a common generalization of many fundamental Morita equivalences that are used to work with induced representations of crossed products defined in Chapter 5. Chapter 6 is devoted to orbit and quasi-orbit spaces and to the Mackey-Glimm dichotomy for orbit spaces. The Takai duality theorem, coincidence of the full and reduced crossed product for amenable groups, twisted crossed products and some structural results comprise Chapter 7. The ideal structure of crossed products is discussed in Chapter 8 and the Gootman-Rosenberg-Sauvageot theorem used here is proved in the final Chapter. Nine appendices contain supplementary results on amenability, vector-valued integration, bundles of \(C^*\)-algebras, representations of \(C^*\)-algebras, the Fell topology, etc.

The book begins with an overview of the group theory of locally compact groups and Haar measure. Then dynamical systems and their associated crossed products are defined (Chapter 2), the structure of the group \(C^*\)-algebras of Abelian and compact groups is worked out and some basic tools are developed (Chapter 3). One of the main results of Chapter 4 is Raeburn’s symmetric imprimitivity theorem, which provides a common generalization of many fundamental Morita equivalences that are used to work with induced representations of crossed products defined in Chapter 5. Chapter 6 is devoted to orbit and quasi-orbit spaces and to the Mackey-Glimm dichotomy for orbit spaces. The Takai duality theorem, coincidence of the full and reduced crossed product for amenable groups, twisted crossed products and some structural results comprise Chapter 7. The ideal structure of crossed products is discussed in Chapter 8 and the Gootman-Rosenberg-Sauvageot theorem used here is proved in the final Chapter. Nine appendices contain supplementary results on amenability, vector-valued integration, bundles of \(C^*\)-algebras, representations of \(C^*\)-algebras, the Fell topology, etc.

Reviewer: V. M. Manuilov (Moskva)

##### MSC:

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

46L55 | Noncommutative dynamical systems |

46L05 | General theory of \(C^*\)-algebras |

22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |

22D30 | Induced representations for locally compact groups |

46L45 | Decomposition theory for \(C^*\)-algebras |

54H15 | Transformation groups and semigroups (topological aspects) |