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Zbl 0922.46050
Raeburn, Iain; Williams, Dana P.
Morita equivalence and continuous-trace $C^*$-algebras. (English)
[B] Mathematical Surveys and Monographs. 60. Providence, RI: American Mathematical Society (AMS). xiv, 327 p. \$ 65.00 (1998). ISBN 0-8218-0860-5/hbk

In this beautiful book, the authors give a complete account on the Dixmier-Douady theory of $C^*$-algebras with continuous trace, using the modern language of Morita equivalence of $C^*$-algebras. Complete means that it contains all material needed so that a graduate student with some basic knowledge in $C^*$-algebras can read and (hopefully) enjoy the book. As a result, this book can also be used as a first introduction to the theory of Hilbert $C^*$-modules, Morita equivalence and Mackey's imprimitivity theorem for group representations, some basics in sheaf cohomology (with non-Abelian coefficient groups), the spectrum of a $C^*$-algebra and its Jacobson topology, and tensor products of $C^*$-algebras. Moreover, the book provides a very nice introduction to some recent research work of the authors (and others) on the interplay between the theory of group actions on continuous-trace $C^*$-algebras and algebraic topology.\par By definition, a $C^*$-algebra with continuous trace is a $C^*$-algebra $A$ which has Hausdorff spectrum $X=\widehat A$ and which is ``locally'' Morita equivalent to $C_0(X)$ (note that by some early results of Fell, this definition is equivalent to the more classical definition using trace functions on the spectrum $X$ induced by positive elements of $A$). Continuous-trace algebras are the ``building stones'' of the more general $C^*$-algebras of type I and they play a very important role in the study of group $C^*$-algebras of nilpotent and semisimple Lie groups. If $A$ is separable and stable, it follows from the Brown-Green-Rieffel theorem (which is strongly related to Kasparov's stabilization theorem) that $A$ is isomorphic to the algebra of $C_0$-sections of some locally trivial $C^*$-algebra bundle over $X$ with fibres isomorphic to ${\cal K}(l^2(\bbfN))$, the compact operators on the standard Hilbert space $\ell^2(\bbfN)$. To each such bundle one can naturally associate a cohomology class $\delta(A)\in H^3(X,\bbfZ)$ (Čech-cohomology), and a now classical result of Dixmier and Douady implies that this process gives a classification of separable and stable continuous-trace algebras with fixed spectrum $X$ by $H^3(X,\bbfZ)$. It was later observed by P. Green (unpublished notes), that, in a similar way, one obtains a classification of all Morita equivalence classes of continuous-trace algebras $A$ with given paracompact spectrum $X$. In fact, he shows that the set $\text{Br}(X)$ of $C_0(X)$-Morita-equivalence classes of continuous-trace algebras with spectrum $X$ forms a group (the Brauer group of $X$) with group operation given by taking tensor products over $X$. The map $\text{Br}(X)\to H^3(X,\bbfZ)$ which maps a class $[A]\in \text{Br}(X)$ to the Dixmier-Douady invariant $\delta(A)\in H^3(X,\bbfZ)$ is then an isomorphism of groups.\par The main purpose of this book is to present all details of these classification results together with a detailed study of the automorphism groups of continuous-trace algebras. The main emphasis lies here on Green's more modern point of view, using Morita equivalence rather than stable isomorphism. This provides the necessary background for the study of group actions on continuous-trace algebras and the interplay between $C^*$-dynamics and algebraic topology. In Chapter 7, the authors give a brief introduction into the equivariant theory, where they study the group $\text{Br}_G(X)$ of Morita equivalence classes of dynamical systems $(A,G,\alpha)$, in which the action of the locally compact group $G$ on $A$ induces a fixed $G$-action on the spectrum $X=\widehat A$. Although the results presented in this chapter are not given in full detail (the details of this would easily fill another book), the chapter certainly serves as a starting point to explore the recent literature on equivariant Brauer groups and the study of the ``fine structure of the Mackey machine'' [as proposed by {\it J. Rosenberg} in his survey ``$C^*$-algebras and Mackey's theory of group representations'', Contemp. Math. 167, 151-181 (1994; Zbl 0866.22005)] for groups acting on $C^*$-algebras with continuous trace.
[S.Echterhoff (Münster)]

MSC 2000:: *46L05 General theory of C*-algebras
46-02 Research monographs (functional analysis)
46M20 Methods of algebraic topology in functional analysis
46L55 Noncommutative dynamical systems
22D25 Operator algebras arising from group representations
46L40 Automorphisms of C*-algebras
43A65 Representations of groups, etc. (abstract harmonic analysis)
46H25 Topological modules
46L35 Classifications and factors of C*-algebras

Keywords: Dixmier-Douady theory $C^*$-algebras with continuous trace; fine structure of the Mackey machine; Hilbert $C^*$-modules; Mackey's imprimitivity theorem for group representations; sheaf cohomology; Jacobson topology; tensor products of $C^*$-algebras; Brown-Green-Rieffel Theorem; Kasparov's stabilization theorem; algebra of $C_0$-sections; locally trivial $C^*$-algebra bundle; Dixmier-Douady invariant; $C^*$-dynamics; algebraic topology; equivariant theory; Morita equivalence classes of dynamical systems; equivariant Brauer groups

Citations: Zbl 0866.22005

Cited in: Zbl 1082.46042 Zbl 1066.46045

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