C*-Algebras and Finite-Dimensional Approximations

This monograph is devoted to the study of C*-algebras with a focus on their finite-dimensional approximations. The book is divided into 17 chapters and accompanied by 6 appendices. The first chapter is an introduction. Chapters 2-10 form Part 1 (Basic Theory), which is devoted, in particular, to the study of nuclear and exact C*-algebras. It contains definitions and basic properties of these classes. It also deals with tensor products and other constructions of C*-algebras (including crossed products, free products and C*-algebras associated to groups) and the behaviour of exactness and nuclearity with respect to these operations is studied, including quasidiagonal C*-algebras and local reflexivity for C*-algebras. Part 2 (Special Topics) comprises chapters 11-14. It deals with simple C*-algebras, approximation properties for groups (including Kazhdan's property (T), Haagerup’s property and weak amenability), the weak expectation property, the local lifting property and weakly exact von Neumann algebras. Chapters 15-17 form Part 3 (Applications), containing results on the classification of von Neumann algebras associated to groups, a solution of the Herrero approximation problem and some counterexamples in K-homology and K-theory.

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