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This book provides an introduction to measure theory and functional
analysis suitable for a beginning graduate course, and is based on
notes the author had developed over several years of teaching such
a course. It is unique in placing special emphasis on the separable
setting, which allows for a simultaneously more detailed and more
elementary exposition, and for its rapid progression into advanced
topics in the spectral theory of families of self-adjoint
operators. The author's notion of measurable Hilbert bundles is
used to give the spectral theorem a particularly elegant
formulation not to be found in other textbooks on the subject.
'The book is very well-written by one of the leading figures in the
subject. It is self-contained, includes relevant recent advances
and is enriched by a large number of examples and illustrations. In
addition to the general bibliography, each chapter includes a
section of notes, which details the authorship of the main results,
and provides useful hints for further readings. Undoubtedly, this
edition will be received by researchers with the same success as
the first one.'European Mathematical SocietyThis is the standard
reference on algebras of Lipschitz functions, written by the
leading figure in the field. The second edition includes new
chapters on nonlinear Banach space geometry, differentiability in
metric measure spaces, and quantum metrics. This latest material
reflects the importance of spaces of Lipschitz functions in a
diverse range of current research directions. Every functional
analyst should have some knowledge of this subject.
The book is a research monograph on the notions of truth and
assertibility as they relate to the foundations of mathematics. It
is aimed at a general mathematical and philosophical audience. The
central novelty is an axiomatic treatment of the concept of
assertibility. This provides us with a device that can be used to
handle difficulties that have plagued philosophical logic for over
a century. Two examples relate to Frege's formulation of
second-order logic and Tarski's characterization of truth
predicates for formal languages. Both are widely recognized as
fundamental advances, but both are also seen as being seriously
flawed: Frege's system, as Russell showed, is inconsistent, and
Tarski's definition fails to capture the compositionality of truth.
A formal assertibility predicate can be used to repair both
problems. The repairs are technically interesting and conceptually
compelling. The approach in this book will be of interest not only
for the uses the author has put it to, but also as a flexible tool
that may have many more applications in logic and the foundations
of mathematics.
Ever since Paul Cohen's spectacular use of the forcing concept to
prove the independence of the continuum hypothesis from the
standard axioms of set theory, forcing has been seen by the general
mathematical community as a subject of great intrinsic interest but
one that is technically so forbidding that it is only accessible to
specialists. In the past decade, a series of remarkable solutions
to long-standing problems in C*-algebra using set-theoretic
methods, many achieved by the author and his collaborators, have
generated new interest in this subject. This is the first book
aimed at explaining forcing to general mathematicians. It
simultaneously makes the subject broadly accessible by explaining
it in a clear, simple manner, and surveys advanced applications of
set theory to mainstream topics.
With a unique approach and presenting an array of new and intriguing topics, Mathematical Quantization offers a survey of operator algebras and related structures from the point of view that these objects are quantizations of classical mathematical structures. This approach makes possible, with minimal mathematical detail, a unified treatment of a variety of topics.
Detailed here for the first time, the fundamental idea of mathematical quantization is that sets are replaced by Hilbert spaces. Building on this idea, and most importantly on the fact that scalar-valued functions on a set correspond to operators on a Hilbert space, one can determine quantum analogs of a variety of classical structures. In particular, because topologies and measure classes on a set can be treated in terms of scalar-valued functions, we can transfer these constructions to the quantum realm, giving rise to C*- and von Neumann algebras.
In the first half of the book, the author quickly builds the operator algebra setting. He uses this as a unifying theme in the second half, in which he treats several active research topics, some for the first time in book form. These include the quantum plane and tori, operator spaces, Hilbert modules, Lipschitz algebras, and quantum groups.
For graduate students, Mathematical Quantization offers an ideal introduction to a research area of great current interest. For professionals in operator algebras and functional analysis, it provides a readable tour of the current state of the field.
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