In this chapter we extend the results of Chapter 6 from finite-dimensional ranges Mn to maps with range B(H). We then develop the immediate applications of the extension theorems to dilation theory. We begin with some observations of a general functional-analytic nature.

Let X and Y be Banach spaces, let Y* denote the dual of Y, and let B(X, Y*) denote the bounded linear transformations of X into Y*. We wish to construct a Banach space such that B(X, Y*) is isometrically isomorphic to its dual. This will allow us to endow B(X, Y*) with a weak* topology.

Fix vectors x in X and y in Y, and define a linear functional x ⊗ y on B(X, Y*) by x ⊗ y(L) = L(x)(y). Since |x ⊗ y(L)| ≤ ||L||·||x||·||y||, we see that x ⊗ y is in B(X, Y*)* with ||x ⊗ y|| ≤ ||x|| ||y||. In fact, ||x ⊗ y|| = ||x|| ||y|| (Exercise 7.1).

It is not difficult to check that the above definition is bilinear, i.e., x ⊗ (y1 + y2) = x ⊗ y1 + x ⊗ y2, (x1 + x2) ⊗ y = x1 ⊗ y + x2 ⊗ y, and (λx) ⊗ y = x ⊗ (λy) = λ(x ⊗ y) for λ ∈. We let Z denote the closed linear span in B(X, Y*)* of these elementary tensors. Actually, Z can be identified as the completion of X ⊗ Y with respect to a cross-norm (Exercise 7.1), but we shall not need that fact here.