The nature of C ]*-algebras is such that one cannot study perturbation
without also studying the theory of lifting and the theory of
extenstions. 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 transivity 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.
