Iain Raeburn

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 behavior 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 generalizations which seem particularly promising. The volume is suitable for graduate students and research mathematicians interested in graph theory and operator algebras.

In this text, the authors give a modern treatment of the classification of continuous-trace $C^*$-algebras up to Morita equivalence. This includes a detailed discussion of Morita equivalence of $C^*$-algebras, a review of the necessary sheaf cohomology, and an introduction to recent developments in the area. The book is accessible to students who are beginning research in operator algebras after a standard one-term course in $C^*$-algebras.The authors have included introductions to necessary but nonstandard background. Thus they have developed the general theory of Morita equivalence from the Hilbert module, discussed the spectrum and primitive ideal space of a $C^*$-algebra including many examples, and presented the necessary facts on tensor products of $C^*$-algebras starting from scratch. Motivational material and comments designed to place the theory in a more general context are included. The text is self-contained and would be suitable for an advanced graduate or an independent study course.