$textrm{C}
*$-approximation theory has provided the foundation for many of the most important conceptual breakthroughs and applications of operator algebras. This book systematically studies (most of) the numerous types of approximation properties that have been important in recent years: nuclearity, exactness, quasidiagonality, local reflexivity, and others. Moreover, it contains user-friendly proofs, insofar as that is possible, of many fundamental results that were previously quite hard to extract from the literature. Indeed, perhaps the most important novelty of the first ten chapters is an earnest attempt to explain some fundamental, but difficult and technical, results as painlessly as possible. The latter half of the book presents related topics and applications--written with researchers and advanced, well-trained students in mind. The authors have tried to meet the needs both of students wishing to learn the basics of an important area of research as well as researchers who desire a fairly comprehensive reference for the theory and applications of $textrm{C}
*$-approximation theory.

Fundamental facts
BASIC THEORY

Nuclear and exact $\textrm{C}*$-algebras: Definitions, basic facts and examples

Tensor products

Constructions

Exact groups and related topics

Amenable traces and Kirchberg's factorization property

Quasidiagonal C*-algebras

AF embeddability

Local reflexivity and other tensor product conditions

Summary and open problems
SPECIAL TOPICS

Simple $\textrm{C}*$-algebras

Approximation properties for groups

Weak expectation property and local lifting property

Weakly exact von Neumann algebras
APPLICATIONS

Classification of group von Neumann algebras

Herrero's approximation problem

Counterexamples in $\textrm{K}$-homology and $\textrm{K}$-theory
APPENDICES

Ultrafilters and ultraproducts

Operator spaces, completely bounded maps and duality

Lifting theorems

Positive definite functions, cocycles and Schoenberg's Theorem

Groups and graphs

Bimodules over von Neumann algebras

Bibliography

Notation index

Subject index