The first edition of this book was written almost twenty-five years ago. Since then the theory of trigonometric series has undergone considerable change. It has always been one of the central parts of Analysis, but now we see its notions and methods appearing, in abstract form, in distant fields like the theory of groups, algebra, theory of numbers. These abstract extensions are, however, not considered here and the subject of the second edition of this book is, as before, the classical theory of Fourier series, which may be described as the meeting ground of the Real and Complex Variables.

This theory has been a source of new ideas for analysts during the last two centuries, and is likely to be so in years to come. Many basic notions and results of the theory of functions have been obtained by mathematicians while working on trigonometric series. Conceivably these discoveries might have been made in different contexts, but in fact they came to life in connexion with the theory of trigonometric series. It was not accidental that the notion of function generally accepted now was first formulated in the celebrated memoir of Dirichlet (1837) dealing with the convergence of Fourier series; or that the definition of Riemann's integral in its general form appeared in Riemann's Habilitationsschrift devoted to trigonometric series; or that the theory of sets, one of the most important developments of nineteenth-century mathematics, was created by Cantor in his attempts to solve the problem of the sets of uniqueness for trigonometric series.

In many parts of analysis an important role is played by multilinear maps. Recall that if E, F, and Z are vector spaces, then a map γ: E × F → Z is bilinear provided that it is linear in each variable, i.e., γ(e1 + e2, f1) = γ(e1, f1) + γ(e2, f1), γ(e1, f1 + f2) = γ(e1, f1) + γ(e1, f2), and γ(λe1, f1) = γ(e1, λf1) = λγ(e1, f1) for any e1 and e2 in E, for any f1 and f2 in F, and for any λ in. If one forms the algebraic tensor product E ⊗ F of E and F, then there is a one-to-one correspondence between linear maps Г: E ⊗ F → Z and bilinear maps γ: E × F → Z given by setting Г(e ⊗ f) = γ(e, f).

Consequently, if one endows E ⊗ F with a matrix norm, then the completely bounded linear maps from E ⊗ F to another matrix-normed space Z correspond to a family of bilinear maps from E × F to Z that one would like to regard as the “completely bounded” bilinear maps. In this fashion, one often arrives at an important family of bilinear maps to study.

The Gelfand–Naimark–Segal theorem gives an abstract characterization of the Banach ∗-algebras that can be represented ∗-isomorphically as C*-subalgebras of B(H) for some Hilbert space H. Thus, the GNS theorem frees us from always having to regard C*-algebras as concrete subalgebras of some Hilbert space. At the same time we may continue to regard them as concrete C*-subalgebras when that might aid us in a proof. For example, proving that the quotient of a concrete C*-subalgebra of some B(H) by a two-sided ideal can again be regarded as a C*-subalgebra of some B(K) would be quite difficult without the GNS theorem. On the other hand defining the norm on Mn(A) and many other constructions are made considerably easier by regarding A as a concrete C*-subalgebra of some B(H).

In this chapter we shall develop the Choi–Effros [49] abstract characterization of operator systems and Ruan's [203] abstract characterization of operator spaces. In analogy with the GNS theory, these characterizations will free us from being forced to regard operator spaces and systems as concrete subspaces of operators.

We begin with the theory of abstract operator systems. We wish to characterize operator systems up to complete order isomorphism. To this end, let S be a complex vector space, and assume that there exists a conjugate linear map s → s* on S with (s*)* = s for all s in S.

Surely, Antoni Zygmund's Trigonometric Series has been, and continues to be, one of the most influential books in the history of mathematical analysis. Therefore, the current printing, which ensures the future availability of this work to the mathematical public, is an event of major importance. Its tremendous longevity is a testimony to its depth and clarity. Generations of mathematicians from Hardy and Littlewood to recent classes of graduate students specializing in analysis have viewed Trigonometric Series with enormous admiration and have profited greatly from reading it. In light of the magnitude of Antoni Zygmund as a mathematician and of the impact of Trigonometric Series, it is only fitting that a brief discussion of his life and mathematics accompany the present volume, and this is what I have attempted to give here. I can only hope that it provides at least a small glimpse into the story of this masterpiece and of the man who produced it.

Antoni Zygmund was born on December 26, 1900 in Warsaw, Poland. His parents had received relatively little education, and were of modest means, so that his background was far less privileged than that of the vast majority of his colleagues. Zygmund attended school through the middle of high school in Warsaw. When World War I broke out, his family was evacuated to Poltava in the Ukraine, where he continued his studies. When the war ended in 1918, his family returned to Warsaw, where he completed pre-collegiate work, and entered Warsaw University.

In the last chapter we saw how the abstract characterization of operator algebras led to a number of factorization formulas for certain universal operator algebras. However, this theory was an isometric theory. In this chapter we focus on the isomorphic theory of operator algebras and applications to similarity questions.

We present Pisier's remarkable work on similarity degree and factorization degree, and Blecher's characterization of operator algebras up to cb isomorphism.

Pisier's work shows that for an operator algebra B, every bounded homomorphism is completely bounded if and only if the type of factorization occurring in the study of MAXA(B) can be carried out with uniform control on the number of factors needed. The least such integer is the factorization degree of the algebra.

Pisier's work has a number of deep implications in the study of bounded representations of groups and in the study of Kadison's similarity conjecture. We focus primarily on Kadison's conjecture, that every bounded homomorphism of a C*-algebra into B(H) is similar to a ∗-homomorphism. Thus, we will show that Kadison's conjecture is equivalent to the existence of an integer d such that every C*-algebra has factorization degree at most d.

A pivotal role in Pisier's work is played by the universal operator algebra of an operator space.

In this chapter we apply the results of Chapter 9 to the study of multiply connected K-spectral sets. We show that for a “nice” region X with finitely many holes it is possible to write down a fairly simple characterization of the family of operators that, up to similarity, have normal ∂X-dilations. This constitutes a model theory for these operators. In contrast, if X has two or more holes, then it is still an open problem to determine whether or not every operator for which X is a spectral set has a normal ∂X-dilation, i.e., is a complete spectral set. A further difficulty with the theory of spectral sets is that it is quite difficult to determine if a given set is a spectral set for an operator. We will illustrate this difficulty in the case that X is an annulus and T is a 2 × 2 matrix.

Thus, even if it is eventually determined that the properties of being a spectral set and being a complete spectral set are equivalent, the use of the theory might be limited by the impossibility of recognizing operators to which it could be applied.

It is easier to determine when a “nice” set with no holes is a spectral set for an operator.

Polynomially bounded and power-bounded operators have played an important role in the development of this area, and there are a number of interesting results, counterexamples, and open questions about these operators. In particular, we will present Foguel's example [98] of a power-bounded operator and Pisier's example [191] of a polynomially bounded operator that are not similar to contractions.

Recall that an operator T is power-bounded provided that there is a constant M such that ||T n|| ≤ M for all n ≥ 0. Clearly, if T = S-1CS with C a contraction, then T is power-bounded with ||T n|| ≤ ||S-1|| ||S||.

It is fairly easy to see (Exercise 10.1), by using the Jordan form, that a matrix T ∈ Mn is power-bounded if and only if it is similar to a contraction. Sz.-Nagy [229] proved that the same characterization holds when T is a compact operator. This led naturally to the conjecture that an arbitrary operator is similar to a contraction if and only if it is power-bounded. Foguel provided the first example of a power-bounded operator that is not similar to a contraction.

Recall that an operator is polynomially bounded provided there is a constant K such that ||p(T)|| ≤ K||p||∞ for every polynomial p, where the ∞-norm is the supremum norm over the unit disk.