This book is intended to give the reader an introduction to the principal results and ideas in the theories of completely positive maps, completely bounded maps, dilation theory, operator spaces, and operator algebras, together with some of their main applications. It is intended to be self-contained and accessible to any reader who has had a first course in functional analysis that included an introduction to C*-algebras. It could be used as a text for a course or for independent reading. With this in mind, we have included plenty of exercises.

We have made no attempt at giving a full state-of-the-art exposition of any of these fields. Instead, we have tried to give the reader an introduction to many of the important techniques and results of these fields, together with a feel for their connections and some of the important applications of the ideas. However, we present new proofs and approaches to some of the well-known results in this area, which should make this book of interest to the expert in this area as well as to the beginner.

The quickest route to a result is often not the most illuminating. Consequently, we occasionally present more than one proof of some results. For example, scattered throughout the text and exercises are five different proofs of a key inequality of von Neumann. We feel that such redundancy can lead to a deeper understanding of the material.