This book focuses on the major applications of martingales to the geometry of Banach spaces, and a substantial discussion of harmonic analysis in Banach space valued Hardy spaces is also presented. It covers exciting links between super-reflexivity and some metric spaces related to computer science, as well as an outline of the recently developed theory of non-commutative martingales, which has natural connections with quantum physics and quantum information theory. Requiring few prerequisites and providing fully detailed proofs for the main results, this self-contained study is accessible to graduate students with a basic knowledge of real and complex analysis and functional analysis. Chapters can be read independently, with each building from the introductory notes, and the diversity of topics included also means this book can serve as the basis for a variety of graduate courses.

These notes revolve around three similarity problems, appearing in three different contexts, but all dealing with the space B(H) of all bounded operators on a complex Hilbert space H. The first one deals with group representations, the second one with C* -algebras and the third one with the disc algebra. We describe them in detail in the introduction which follows. This volume is devoted to the background necessary to understand these three problems, to the solutions that are known in some special cases and to numerous related concepts, results, counterexamples or extensions which their investigation has generated. While the three problems seem different, it is possible to place them in a common framework using the key concept of "complete boundedness," which we present in detail. Using this notion, the three problems can all be formulated as asking whether "boundedness" implies "complete boundedness" for linear maps satisfying certain additional algebraic identities.

Based on the author's university lecture courses, this book presents the many facets of one of the most important open problems in operator algebra theory. Central to this book is the proof of the equivalence of the various forms of the problem, including forms involving C*-algebra tensor products and free groups, ultraproducts of von Neumann algebras, and quantum information theory. The reader is guided through a number of results (some of them previously unpublished) revolving around tensor products of C*-algebras and operator spaces, which are reminiscent of Grothendieck's famous Banach space theory work. The detailed style of the book and the inclusion of background information make it easily accessible for beginning researchers, Ph.D. students, and non-specialists alike.

In this book the authors give the first necessary and sufficient conditions for the uniform convergence a.s. of random Fourier series on locally compact Abelian groups and on compact non-Abelian groups. They also obtain many related results. For example, whenever a random Fourier series converges uniformly a.s. it also satisfies the central limit theorem. The methods developed are used to study some questions in harmonic analysis that are not intrinsically random. For example, a new characterization of Sidon sets is derived. The major results depend heavily on the Dudley-Fernique necessary and sufficient condition for the continuity of stationary Gaussian processes and on recent work on sums of independent Banach space valued random variables. It is noteworthy that the proofs for the Abelian case immediately extend to the non-Abelian case once the proper definition of random Fourier series is made. In doing this the authors obtain new results on sums of independent random matrices with elements in a Banach space. The final chapter of the book suggests several directions for further research.

The first part of this book is an introduction with emphasis on examples that illustrate the theory of operator spaces. The second part is devoted to applications to C*-algebras, with a systematic exposition of tensor products of C* algebras. The third part of the book describes applications to non self-adjoint operator algebras and similarity problems. The author's counterexample to the "Halmos problem" is presented, along with work on the new concept of "length" of an operator algebra.

Now in paperback, this popular book gives a self-contained presentation of a number of recent results, which relate the volume of convex bodies in n-dimensional Euclidean space and the geometry of the corresponding finite-dimensional normed spaces. The methods employ classical ideas from the theory of convex sets, probability theory, approximation theory, and the local theory of Banach spaces. The first part of the book presents self-contained proofs of the quotient of the subspace theorem, the inverse Santalo inequality and the inverse Brunn-Minkowski inequality. In the second part Pisier gives a detailed exposition of the recently introduced classes of Banach spaces of weak cotype 2 or weak type 2, and the intersection of the classes (weak Hilbert space). This text will be a superb choice for courses in analysis and probability theory.