It is assumed throughout this book that the reader is familiar with operator theory and the basic properties of C*-algebras (see for example [76] and [8, Chapter 1]). We concentrate primarily on giving a self-contained exposition of the theory of completely positive and completely bounded maps between C*-algebras and the applications of these maps to the study of operator algebras, similarity questions, and dilation theory. In particular, we assume that the reader is familiar with the material necessary for the Gelfand–Naimark–Segal theorem, which states that every C*-algebra has a one-to-one, ∗-preserving, norm-preserving representation as a norm-closed, ∗-closed algebra of operators on a Hilbert space.

In this chapter we introduce some of the key concepts that will be studied in this book.

As well as having a norm, a C*-algebra also has an order structure, induced by the cone of positive elements. Recall that an element of a C*-algebra is positive if and only if it is self-adjoint and its spectrum is contained in the nonnegative reals, or equivalently, if it is of the form a*a for some element a. Since the property of being positive is preserved by ∗-isomorphism, if a C*-algebra is represented as an algebra of operators on a Hilbert space, then the positive elements of the C*-algebra coincide with the positive operators that are contained in the representation of the algebra.