COMPLETELY BOUNDED MAPS AND OPERATOR ALGEBRAS (Cambridge Studies in Advanced Mathematics 78) By VERN PAULSEN: 300 pp., 47.50 (US$65.00), ISBN 0-521-81669-6 (Cambridge University Press, 2002)

operator algebras are the focus of the second half of the book. Such an algebra may be naturally defined as an abstract operator space carrying a completely contractive multiplication. However, it was not known for some time whether such an abstract operator algebra can be realised as a concrete one; that is, whether it can be completely isometrically embedded into B(H) for some Hilbert space H. The solution to this problem (in the unital case) is given by the Blecher–Ruan–Sinclair theorem, which was obtained in 1990. It provides us with the non-selfadjoint analogue of the celebrated Gelfand–Naimark–Segal theorem for C∗-algebras. Two approaches to this fundamental result are offered in the book under review. In Chapter 16 a novel route is taken, which rests on the concept of an injective envelope, originally devised by Hamana in the late 1970s and explained and examined in a new way in Chapter 15. Amongst others, this approach has the advantage that an extension of the Christensen–Effros–Sinclair representation theorem for operator bimodules is obtained at the same time. A more traditional route to the BRS theorem is taken in Chapter 17. Multiplication is a bilinear operation, and is thus naturally related to tensor products. The tensor product that turns out to be best-suited to completely bounded multilinear maps is the Haagerup tensor product; hence a careful examination of its properties, as well as its relation to free products of C∗-algebras, forms part of this chapter. Another proof of the BRS theorem is then presented in Theorem 17.9. Chapter 18 is devoted to several interesting applications of the BRS theorem, which we have no space to discuss here. Of course, a representation of operator algebras as concrete non-selfadjoint algebras of operators on Hilbert space first needed an abstract characterisation of (concrete) operator spaces. This fundamental result belongs to Ruan; it was the starting point of non-commutative functional analysis, the theory of operator spaces and completely bounded operators. Chapter 13 contains an exposition of Ruan’s theorem and its ‘order-theoretic brother’, the abstract characterisation of operator systems (the canonical domains of completely positive operators) due to Choi and Effros. In Chapter 14, entitled ‘An operator space bestiary’, the author takes a closer look at a number of important examples of, and constructions with, operator spaces – thus completing the essential toolkit of the theory. One of the striking successes of non-commutative functional analysis was Pisier’s solution to Halmos’s similarity problem, a question in ‘traditional’ Hilbert space operator theory. This solution is presented in detail in Chapter 10, which follows work by Davidson and Paulsen from 1997. An operator T ∈B(H) is said to be power-bounded if there is a constant M such that ‖T‖ M for all n ∈ N. Clearly, every operator that is similar to a contraction must be power-bounded. The conjecture that power-boundedness is also sufficient for similarity to a contraction was shown to be false by Foguel in 1964. Halmos asked in 1970 whether the stronger condition ‖p(T )‖ K‖p‖∞ for every polynomial p (where the supremum norm is taken over the unit disk in the complex plane) is sufficient; such operators T are called polynomially bounded. This problem remained open for more than 25 years, until Pisier produced a counterexample in 1997. An important step forward had been made by Paulsen in 1984, by showing that T is similar to a contraction if and only if it is completely polynomially bounded; that is, the canonical representation of the disk algebra A(D) on H is completely bounded.