Dana P. Williams

Iain Raeburn

A modern treatment of this complex mathematical topic for students beginning research in operator algebras as well as mathematical physicists. Topics include the algebra of compact operators, sheaves, cohomology, the Brauer group and group actions, and the imprimivity theorem. The authors assume a knowledge of C*-algebras, the Gelfand-Naimark Theorem, continuous functional calculus, positivity, and the GNS- construction. Annotation copyrighted by Book News, Inc., Portland, OR

Morita Equivalence and Continuous-Trace $C^*$-Algebras

Siegfried Echterhoff

S. Kaliszewski

John Quigg

Imprimitivity theorems provide a fundamental tool for studying the representation theory and structure of crossed-product $C^*$-algebras. In this work, we show that the Imprimitivity Theorem for induced algebras, Green's Imprimitivity Theorem for actions of groups, and Mansfield's Imprimitivity Theorem for coactions of groups can all be viewed as natural equivalences between various crossed-product functors among certain equivariant categories. The categories involved have $C^*$-algebras with actions or coactions (or both) of a fixed locally compact group $G$ as their objects, and equivariant equivalence classes of right-Hilbert bimodules as their morphisms. Composition is given by the balanced tensor product of bimodules. The functors involved arise from taking crossed products; restricting, inflating, and decomposing actions and coactions; inducing actions; and various combinations of these. Several applications of this categorical approach are also presented, including some intriguing relationships between the Green and Mansfield bimodules, and between restriction and induction of representations.

A Categorical Approach to Imprimitivity Theorems for $C^*$-Dynamical Systems

Graph algebras are a family of operator algebras which are associated to directed graphs. These algebras have an attractive structure theory in which algebraic properties of the algebra are related to the behaviour of paths in the underlying graph. In the past few years there has been a great deal of activity in this area, and graph algebras have cropped up in a surprising variety of situations, including non-abelian duality, non-commutative geometry, and the classification of simple $C^*$-algebras. The first part of the book provides an introduction to the subject suitable for students who have seen a first course on the basics of $C^*$-algebras. In the second part, the author surveys the literature on the structure theory of graph algebras, highlights some applications of this theory, and discusses several recent generalisations which seem particularly promising.

Iain Raeburn

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