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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.
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.
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.
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.
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.
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.
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