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\(K\)-theory for group \(C^*\)-algebras and semigroup \(C^*\)-algebras. (English) Zbl 1390.46001

Oberwolfach Seminars 47. Cham: Birkhäuser/Springer (ISBN 978-3-319-59914-4/pbk; 978-3-319-59915-1/ebook). ix, 319 p. (2017).
This nicely written book focuses on \(C^*\)-algebras attached to groups and semigroups, and on their \(K\)-theory computations, which often allow to determine their structure. The book is written by leading experts on \(C^*\)-algebras, \(C^*\)-dynamical systems and \(K\)-theory. It grew out from the Oberwolfach Seminar with the same title, and can be viewed as a second course on \(C^*\)-algebra theory.
The first half of the book (Chapters 2–4) is a textbook on group \(C^*\)-algebras and on computational aspects of \(K\)-theory. The second half of the book (Chapters 5–7) passes from group \(C^*\)-algebras to semigroup ones and leads the reader from basic definitions to the current state of the art, including some quite recent results.
Chapter 2 starts with the basics of the crossed product theory: \(C^*\)-dynamical systems, Morita equivalence, Fell topology, induced representations, and finishes with the Mackey-Rieffel-Green machine, which allows to study the ideal structure of crossed products.
Chapter 3 gives a brief introduction to bivariant \(KK\)-theory, with the emphasis on the Baum-Connes conjecture, which (if it holds true) provides a bridge to better developed computation methods from algebraic topology.
Chapter 4 deals with the quantitative \(K\)-theory based on geometric structures on \(C^*\)-algebras, which allows to use the cutting-and-pasting technique in \(K\)-theory computations.
Chapter 5 introduces the theory of partial dynamical systems and semigroup \(C^*\)-algebras. Semigroup \(C^*\)-algebras are more numerous than group \(C^*\)-algebras and often enjoy quite different properties. Two important conditions (independence and Toeplitz) are introduced, which allow to study the structure of semigroup \(C^*\)-algebras, and interesting classes of examples coming from group theory and number theory are discussed.
The book ends with more examples of semigroup \(C^*\)-algebras. In Chapter 6, the \(C^*\)-algebras associated with an algebraic number field are considered, and in Chapter 7, the semigroup \(C^*\)-algebras for finitely generated subsemigroups of \(\mathbb Z^n\) (i.e., \(C^*\)-algebras of toric varieties) are studied, and their \(K\)-theory is computed for \(n=2\).

MSC:

46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
46L80 \(K\)-theory and operator algebras (including cyclic theory)
46L05 General theory of \(C^*\)-algebras
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