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.