The nature of $C^*$-algebras is such that one cannot study perturbation without also studying the theory of lifting and the theory of extensions. Approximation questions involving representations of relations in matrices and $C^*$-algebras are the central focus of this volume. A variety of approximation techniques are unified by translating them into lifting problems: from classical questions about transitivity of algebras of operators on Hilbert spaces to recent results in linear algebra. One chapter is devoted to Lin's theorem on approximating almost normal matrices by normal matrices.The techniques of universal algebra are applied to the category of $C^*$-algebras. An important difference, central to this book, is that one can consider approximate representations of relations and approximately commuting diagrams. Moreover, the highly algebraic approach does not exclude applications to very geometric $C^*$-algebras. $K$-theory is avoided, but universal properties and stability properties of specific $C^*$-algebras that have applications to $K$-theory are considered. Index theory arises naturally, and very concretely, as an obstruction to stability for almost commuting matrices. Multiplier algebras are studied in detail, both in the setting of rings and of $C^*$-algebras. Recent results about extensions of $C^*$-algebras are discussed, including a result linking amalgamated products with the Busby/Hochshild theory.
