In the last chapter we obtained an abstract characterization of operator spaces that allows us to define these spaces without a concrete representation. This result has had a tremendous impact and has led to the development of a general theory of operator spaces that parallels in some ways the development of the theory of Banach spaces.

In this chapter we give the reader a brief introduction to some of the basics of this theory, focusing on some of the more important operator spaces that we will encounter in later chapters.

We have already encountered one example of the power of this axiomatic characterization. In Exercise 13.3, it was shown that if V is an operator space and W ⊆ V a closed subspace, then V / W is an operator space, where the matrix norm structure on V/W comes from the identification Mm,n(V/W) = Mm,n(V)/Mm,n(W). Yet in most concrete situations it is difficult to actually exhibit a concrete completely isometric representation of V/W as operators on a Hilbert space.

The first natural question in the area is as follows: If V is, initially, just a normed space, then is it always possible to assign norms ||·||m,n to Mm,n(V) for all m and n in such a fashion that V becomes an operator space? The answer to this question is yes.

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.

In this chapter we take a closer look at injectivity and introduce injective envelopes and C*-envelopes of operator systems, operator algebras, and operator spaces. Loosely speaking, the injective envelope of an object is a “minimal” injective object that contains the original object. The C*-envelope of an operator algebra is a generalization of the Silov boundary of a uniform algebra. The C*-envelope of an operator algebra A is the “smallest” C*-algebra that contains A as a subalgebra, up to completely isometric isomorphism. These ideas will be made precise in this chapter. Many of the ideas of this chapter are derived from the work of M. Hamana [112].

Injectivity is really a categorical concept. Suppose that we are given some category C consisting of objects and morphisms. Then an object I is called injective in C provided that for every pair of objects E ⊆ F and every morphism φ: E → I, there exists a morphism ψ: F → I that extends φ, i.e., such that ψ(e) = φ(e) for every e in E.

If we let denote the collection of operator systems and define the morphisms between operator systems to be the completely positive maps, then since the composition of completely positive maps is again completely positive, we shall have a category, which we call the category of operator systems.