This volume expounds the general theory of Banach algebras and shows how their topology is often determined by their algebraic structures. It synthesizes work in this area from the past 20 years, including many new and unpublished results, and probes the questions of when homomorphisms and derivations from Banach algebras are automatically continuous and seeks canonical forms for these maps. Many specific classes of Banach algebras are described, including function algebras, group algebras, algebras of perators, C*-algebras, and radical Banach algebras. This is essential reading for anyone...

Profinite groups are of interest to mathematicians working in a variety of areas, including number theory, abstract groups, and analysis. The underlying theory reflects these diverse influences, with methods drawn from both algebra and topology and with fascinating connections to field theory. This is the first book to be dedicated solely to the study of general profinite groups. It provides a thorough introduction to the subject, designed not only to convey the basic facts but also to enable readers to enhance their skills in manipulating profinite groups. The first few chapters lay the...

This book examines the number-theoretic properties of the real numbers. It collects a variety of new ideas and develops connections between different branches of mathematics. An indispensable compendium of basic results, the text also includes important theorems and open problems. The book begins with the classical results of Borel, Khintchine, and Weyl, and then proceeds to Diophantine approximation, GCD sums, Schmidt's method, and uniform distribution. Other topics include generalizations to higher dimensions and various non-periodic problems (for example, restricting approximation to...

Finite Coxeter groups and related structures arise naturally in several branches of mathematics such as the theory of Lie algebras and algebraic groups. The corresponding Iwahori-Hecke algebras are then obtained by a certain deformation process which have applications in the representation theory of groups of Lie type and the theory of knots and links. This book develops the theory of conjugacy classes and irreducible character, both for finite Coxeter groups and the associated Iwahori-Hecke algebras. Topics covered range from classical results to more recent developments and are clear and...

A revised and expanded second edition of Reiter's classic text Classical Harmonic Analysis and Locally Compact Groups (Clarendon Press 1968). It deals with various developments in analysis centring around around the fundamental work of Wiener, Carleman, and especially A. Weil. It starts with the classical theory of Fourier transforms in euclidean space, continues with a study at certain general function algebras, and then discusses functions defined on locally compact groups. The aim is, firstly, to bring out clearly the relations between classical analysis and group theory, and secondly, to...

This book is an indispensable source for anyone with an interest in semigroup theory or whose research overlaps with this increasingly important and active field of mathematics. It clearly emphasizes "pure" semigroup theory, in particular the various classes of regular semigroups. More than 150 exercises, accompanied by relevant references to the literature, give pointers to areas of the subject not explicitly covered in the text.

Modern local spectral theory is built on the classical spectral theorem, a fundamental result in single-operator theory and Hilbert spaces. This book provides an in-depth introduction to the natural expansion of this fascinating topic of Banach space operator theory. It gives complete coverage of the field, including the fundamental recent work by Albrecht and Eschmeier which provides the full duality theory for Banach space operators. One of its highlights are the many characterizations of decomposable operators, and of other related, important classes of operators, including identifications...

In analytic number theory many problems can be "reduced" to those involving the estimation of exponential sums in one or several variables. This book is a thorough treatment of the developments arising from the method for estimating the Riemann zeta function. Huxley and his coworkers have taken this method and vastly extended and improved it. The powerful techniques presented here go considerably beyond older methods for estimating exponential sums such as van de Corput's method. The potential for the method is far from being exhausted, and there is considerable motivation for other...

This text presents easy to understand proofs of some of the most difficult results about polynomials. It encompasses a self-contained account of the properties of polynomials as analytic functions of a special kind.
The zeros of compositions of polynomials are also investigated along with their growth, and some of these considerations lead to the study of analogous questions for trigonometric polynomials and certain transcendental entire functions. The strength of methods are fully explained and demonstrated by means of applications.

In this volume, the authors introduce the theory of existentially closed groups, bringing together both well-established and more contemporary ideas, interpretations, and proofs. They adopt a group-theoretical rather than a model-theoretical point of view as they define existentially closed groups and summarizes some of the techniques that are basic to infinite group theory, such as the formation of free products with amalgamation and HNN-extensions. From this basis the theory is developed and many of the more recently discovered results are proved and discussed.