London Mathematical Society Monographs – serie

Banach algebras combine algebraic and analytical aspects: it is the interplay of these structures that gives the subject its fascination. This volume expounds the general theory of Banach algebras, and shows how their topology is often determined by their algebraic structure: the central questions ask when homomorphisms and derivations from Banach algebras are automatically continuous, and seek canonical forms for these maps. The book synthesizes work over the last 20 years, and gives a definitive account; there are many new and unpublished results. The book describes many specific classes of Banach algebras, including function algebras, group algebras, algebras of operators, C*-algebras, and radical Banach algebras; it is a compendium of results on these examples. The subject interweaves algebra, functional analysis, and complex analysis, and has a dash of set theory and logic; the background in all these areas is fully explained. This volume is essential reading for anyone interested in any aspect of this vast subject.

This book is concerned with applications of functional analysis and complex analysis to problems of interpolation in spaces of analytic functions. The problems we look at are those of recovery, producing approximations to functions from measured values. These values may in turn be corrupted by small errors and we wish to be able to produce a good model using this partial and inaccurate information. The practical applications include systems identification, signal processing, and sampling. A selection of the material of this book would be appropriate for a graduate course on function spaces and operators acting on them. Chapter 8 gives a mathematician's introduction to H( control theory, one of the big research areas of the last 15 years. Worst-case identification (discussed in Chapters 3,4, and 6) is a major area of modern systems theory to which the author has made many contributions. This book gives the first theoretical treatment of this area: it includes much practical material on input design and identification algorithms. Sampling and systems processing is another active area of research. The book presents an accessible treatment of several advanced topics, some included for the first time in any book.

This is the first book to be dedicated entirely to profinite groups, an area of algebra with important links to number theory and other areas of mathematics. It provides a comprehensive overview of the subject; prerequisite knowledge is kept to a minimum, and several major theorems are presented in an accessible form. The book would provide a valuable introduction for postgraduate students, or form a useful reference for researchers in other areas. The first few chapters lay the foundations and explain the role of profinite groups in number theory. Later chapters explore various aspects of profinite groups in more detail; these contain accessible and lucid accounts of many major theorems. Prerequisites are kept to a minimum with the basic topological theory summarized in an introductory chapter.

This book deals with the number-theoretic properties of almost all real numbers. It brings together many different types of result never covered within the same volume before, thus showing interactions and common ideas between different branches of the subject. It provides an indispensable compendium of basic results, important theorems and open problems. Starting from the classical results of Borel, Khintchine and Weyl, normal numbers, Diophantine approximation and uniform distribution are all discussed. Questions are generalized to higher dimensions and various non-periodic problems are also considered (for example restricting approximation to fractions with prime numerator and denominator). Finally, the dimensions of some of the exceptional sets of measure zero are considered.

Finite Coxeter groups and related structures arise naturally in several branches of mathematics, for example, the theory of Lie algebras and algebraic groups. The corresponding Iwahori-Hecke algebras are obtained by a certain deformation process. They have applications in the representation theory of groups of Lie type and the theory of knots and links. The aim of this book is to develop the theory of conjugacy classes and irreducible characters, both for finite Coxeter groups and the associated Iwahori-Hecke algebras. The topics range from classical results to more recent developments and are treated in a coherent and self-contained way. This is the first book which develops these subjects both from a theoretical and an algorithmic point of view in a systematic way. All types of finite Coxeter groups are covered.

In this important and original monograph, useful for both academic and professional researchers and students of mathematics and physics, the author describes his work on the Riemann zeta function and its adelic interpretation. It provides an original point of view, bringing new, highly useful dictionaries between different fields of mathematics. It develops an arithmetical approach to the continuum of real numbers and unifies many areas of mathematics including: Markov Chains, q-series, Elliptic curves, the Heisenberg group, quantum groups, and special functions (such as the Gamma, Beta, Zeta, theta, Bessel functions, the Askey-Wilson and the classical orthagonal polynomials) The text discusses real numbers from a p-adic point of view, first mooted by Araeklov. It includes original work on coherent theory, with implications for number theory and uses ideas from probability theory including Markov chains and noncommutative geometry which unifies the p-adic theory and the real theory by constructing a theory of quantum orthagonal polynomials.

For mathematicians working in group theory, the study of the many infinite-dimensional groups has been carried out in an individual and non-coherent way. For the first time, these apparently disparate groups have been placed together, in order to construct the `big picture'. This book successfully gives an account of this - and shows how such seemingly dissimilar types such as the various groups of operators on Hilbert spaces, or current groups are shown to belong to a bigger entitity. This is a ground-breaking text will be important reading for advanced undergraduate and graduate mathematicians.

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 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 of 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 study basic properties of functions on abelian and non-abelian groups. The book gives a systematic introduction to these topics and endeavours to provide tools for further research. In the new edition relevant material is added that was not yet available at the time of the first edition.

This book is an indispensable source for anyone with an interest in semigroup theory or whose research overlaps with this increasingly important area of mathematics. It is a clear and readable introduction to the subject, with emphasis on various classes of regular and semigroups. More than 150 exercises, accompanied by relevant references to the literature,give pointerse 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, whose pioneers include Dunford, Bishop, Foias, and others. Assuming only modest prerequisites of its readership, 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. It is highlighted by many characterizations of decomposable operators, and of other related, important classes of operators, as well as an in-depth study of their spectral properties, including identifications of distinguished parts, and results on permanence properties of spectra with respect to several types of similarity. Also found is a thorough and quite elementary treatment of the modern single- operator duality theory; this theory has many applications, both to general issues of classification and to such celebrated problems as the invariant subspace problems. A long chapter - almost a book in itself - is devoted to the use of local spectral theory in the study of spectral properties of multipliers and convolution operators. Another one describes its connections to automatic continuity theory. Written in a careful and detailed style, it contains numerous examples, many simplified proofs of classical results, and extensive references. It concludes with a list of interesting open problems, suitable for continued research.

This invaluable reference is the first to present the general theory of algebras of operators on a Hilbert space, and the modules over such algebras. The new theory of operator spaces is presented early on and the text assembles the basic concepts, theory and methodologies needed to equip a beginning researcher in this area.A major trend in modern mathematics, inspired largely by physics, is toward `noncommutative' or `quantized' phenomena. In functional analysis, this has appeared notably under the name of `operator spaces', which is a variant of Banach spaces which is particularly appropriate for solving problems concerning spaces or algebras of operators on Hilbert space arising in 'noncommutative mathematics'.The category of operator spaces includes operator algebras, selfadjoint (that is, C*-algebras) or otherwise. Also, most of the important modules over operator algebras are operator spaces. A common treatment of the subjects of C*-algebras, nonselfadjoint operator algebras, and modules over such algebras (such as Hilbert C*-modules), together under the umbrella of operator space theory, is the main topic of the book. A general theory of operator algebras, and their modules, naturally develops out of the operator space methodology. Indeed, operator space theory is a sensitive enough medium to reflect accurately many important noncommutative phenomena. Using recent advances in the field, the book shows how the underlying operator space structure captures, very precisely, the profound relations between the algebraic and the functional analytic structures involved. The rich interplay between spectral theory, operator theory, C*-algebra and von Neumann algebra techniques, and the influx of important ideas from related disciplines, such as pure algebra, Banach space theory, Banach algebras, and abstract function theory is highlighted. Each chapter ends with a lengthy section of notes containing a wealth of additional information.

This is a reissue of a classic text, which includes the author's own corrections and provides a very accessible, self contained introduction to the classical theory of orders and maximal orders over a Dedekind ring. It starts with a long chapter that provides the algebraic prerequisites for this theory, covering basic material on Dedekind domains, localizations and completions, as well as semisimple rings and separable algebras. This is followed by an introduction to the basic tools in studying orders, such as reduced norms and traces, discriminants, and localization of orders. The theory of maximal orders is then developed in the local case, first in a complete setting, and then over any discrete valuation ring. This paves the way to a chapter on the ideal theory in global maximal orders, with detailed expositions on ideal classes, the Jordan-Zassenhaus Theorem, and genera. This is followed by a chapter on Brauer groups and crossed product algebras, where Hasse's theory of cyclic algebras over local fields is presented in a clear and self-contained fashion. Assuming a couple of facts from class field theory, the book goes on to present the theory of simple algebras over global fields, covering in particular Eichler's Theorem on the ideal classes in a maximal order, as well as various results on the KO group and Picard group of orders. The rest of the book is devoted to a discussion of non-maximal orders, with particular emphasis on hereditary orders and group rings. The ideas collected in this book have found important applications in the smooth representation theory of reductive p-adic groups. This text provides a useful introduction to this wide range of topics. It is written at a level suitable for beginning postgraduate students, is highly suited to class teaching and provides a wealth of exercises.

In analytic number theory a large number of problems can be "reduced" to problems involving the estimation of exponential sums in one or several variables. This book is a thorough treatment of the developments arising from the method developed by Bombieri and Iwaniec in 1986 for estimating the Riemann zeta function on the line *s = 1/2. Huxley and his coworkers (mostly Huxley) 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 researchers to try to master this subject. However, anyone currently trying to learn all of this material has the formidable task of wading through numerous papers in the literature. This book simplifies that task by presenting all of the relevant literature and a good part of the background in one package.The audience for the book will be mathematics graduate students and faculties with a research interest in analytic theory; more specifically, those with an interest in exponential sum methods. The book is self-contained; any graduate student with a one semester course in analytic number theory should have a more than sufficient background.

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 anlytic 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.

It has been known for some time that many of the familiar integrable systems of equations are symmetry reductions of self-duality equations on a metric or on a Yang-Mills connection (for example, the Korteweg-de Vries and nonlinear Schrödinger equations are reductions of the self-dual Yang-Mills equation). This book explores in detail the connections between self-duality and integrability, and also the application of twistor techniques to integrable systems. It has two central themes: first, that the symmetries of self-duality equations provide a natural classification scheme for integrable systems; and second that twistor theory provides a uniform geometric framework for the study of B¨ acklund tranformations, the inverse scattering method, and other such general constructions of integrability theory, and that it elucidates the connections between them.

This book provides an introduction to the theory of existentially closed groups, for both graduate students and established mathematicians. It is presented from a group theoretical, rather than a model theoretical, point of view. The recursive function theory that is needed is included in the text.Interest in existentially closed groups first developed in the 1950s. This book brings together a large number of results proved since then, as well as introducing new ideas, interpretations and proofs.The authors begin by defining existentially closed groups, and summarizing some of the techniques that are basic to infinite group theory (e.g. 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.The aim is to assist group theorists to find their way into a corner of their subject which has its own characteristic flavour, but which is recognizably group theory.

The coclass project (1980-1994) provided a new and powerful way to classify finite p-groups. This monograph gives a coherent account of the thinking from which the philosophy that lead to this classification developed. The authors provide a careful summary and explanation of the many and difficult original research papers on the coclass conjecture and the structure theorem, thus elucidating the background research for those new to the area as well as for experienced researchers. The classification philosophy has lead to many new and challenging problems. Amongst those considered are problems about pro-p-groups, the Grigorchuk group, the Nottingham group as well as linear pro-p-groups. Throughout the book the authors have used a wide range of algebraic techniques and have developed from first principles, or from basic and well known results, the cohomology of groups, spectral sequences, and representation theory. This comprehensive and long-awaited survey of the recent and current research on the structure of finite p-groups will be an important reference for all group theorists.

Pluripotential theory is a recently developed non-linear complex counterpart of classical potential theory. Its main area of application is multidimensional complex analysis.The central part of the pluripotential theory is occupied by maximal plurisubharmonic functions and the generalized complex Monge-Ampère operator. The interplay between these two notions provides the focal point of this monograph, which contains an up-to-date account of the developments from the large volume of recent work in this area.The substantial proportion of this monograph devoted to classical properties of subharmonic and plurisubharmonic functions makes the pluripotential theory available for the first time to a wide audience of analysts.

The aim of this book is to give necessary and sufficient conditions for aCoo map to be C0-stable; the aim is achieved in a wide range of dimensions via a detailed study of the geometry and topology of many classes of "generic" singularities.The methods developed for examining the topology and geometry use results from many areas of mathematics - geometric topology, stratification theory, algebraic topology, algebraic geometry, commutative algebra... - and further progress will doubtless be made from the application of deeper results from these areas. Conversely, it is to be hoped that the description of the behaviour of generic singularities will also have interesting consequences for these areas of mathematics, which are those with most interaction with singularity theory. The book describes original research; essentially none of its results has previously appeared elsewhere, either in scientific articles or in books.

Rapid developments in multivariable spectral theory have led to important and fascinating results which also have applications in other mathematical disciplines. In this book, classical results from the cohomology theory of Banach algebras, multidimensional spectral theory, and complex analytic geometry have been freshly interpreted using the language of homological algebra. It has also been used to give in sights into new developments in the spectral theory of linear operators. Various concepts from function theory and complex analytic geometry are drawn together and used to give a new approach to concrete spectral computations. The advantages of this approach are illustrated by a variety of examples, unexpected applications, and conceptually new ideas which should stimulate further research.

Although Lie polynomials first appeared at the turn of the century, there have been many recent developments especially from the point of view of representation theory. This book covers all aspects, with emphasis on the algebraic and combinatorial point of view as well as representation theory.

The first edition of this book provided an account of the restricted Burnside problem making extensive use of Lie ring techniques to provide a uniform treatment of the field. It also included Kostrikin's theorem for groups of prime exponent.The second edition, as well as providing general updating, contains a new chapter on E.I. Zelmanov's highly acclaimed and recent solution to the Restricted Burnside Problem for arbitrary prime-power exponent. This material is currently only available in papers in Russian journals. This proof of Zelmanov's theorem given in the new edition is self contained, and (unlike Zelmanov's original proof) does not rely on the theory of Jordan algebras.

Super-real fields are a class of large totally ordered fields. These fields are larger than the real line. They arise from quotients of the algebra of continuous functions on a compact space by a prime ideal, and generalize the well-known class of ultrapowers, and indeed the continuous ultrapowers. These fields are of interest in their own right and have many surprising applications, both in analysis and logic. The authors introduce some exciting new fields, including a natural generalization of the real line R, and resolve a number of open problems. The book is intended to be accessible to analysts and logicians. After an exposition of the general theory of ordered fields and a careful proof of some classic theorems, including Kaplansky's embedding theorems , the authors establish important new results in Banach algebra theory, non-standard analysis, an model theory.

This is the first comprehensive reference published on heat equations associated with non self-adjoint uniformly elliptic operators. The author provides introductory materials for those unfamiliar with the underlying mathematics and background needed to understand the properties of heat equations. He then treats Lp properties of solutions to a wide class of heat equations that have been developed over the last fifteen years. These primarily concern the interplay of heat equations in functional analysis, spectral theory and mathematical physics. This book addresses new developments and applications of Gaussian upper bounds to spectral theory. In particular, it shows how such bounds can be used in order to prove Lp estimates for heat, Schrodinger, and wave type equations. A significant part of the results have been proved during the last decade. The book will appeal to researchers in applied mathematics and functional analysis, and to graduate students who require an introductory text to sesquilinear form techniques, semigroups generated by second order elliptic operators in divergence form, heat kernel bounds, and their applications.It will also be of value to mathematical physicists. The author supplies readers with several references for the few standard results that are stated without proofs.

This book seeks to describe the rapid development in recent decades of sieve methods able to detect prime numbers. The subject began with Eratosthenes in antiquity, took on new shape with Legendre's form of the sieve, was substantially reworked by Ivan M. Vinogradov and Yuri V. Linnik, but came into its own with Robert C. Vaughan and important contributions from others, notably Roger Heath-Brown and Henryk Iwaniec. Prime-Detecting Sieves breaks new ground by bringing together several different types of problems that have been tackled with modern sieve methods and by discussing the ideas common to each, in particular the use of Type I and Type II information. No other book has undertaken such a systematic treatment of prime-detecting sieves. Among the many topics Glyn Harman covers are primes in short intervals, the greatest prime factor of the sequence of shifted primes, Goldbach numbers in short intervals, the distribution of Gaussian primes, and the recent work of John Friedlander and Iwaniec on primes that are a sum of a square and a fourth power, and Heath-Brown's work on primes represented as a cube plus twice a cube.This book contains much that is accessible to beginning graduate students, yet also provides insights that will benefit established researchers.

Random matrix theory, both as an application and as a theory, has evolved rapidly over the past fifteen years. Log-Gases and Random Matrices gives a comprehensive account of these developments, emphasizing log-gases as a physical picture and heuristic, as well as covering topics such as beta ensembles and Jack polynomials. Peter Forrester presents an encyclopedic development of log-gases and random matrices viewed as examples of integrable or exactly solvable systems. Forrester develops not only the application and theory of Gaussian and circular ensembles of classical random matrix theory, but also of the Laguerre and Jacobi ensembles, and their beta extensions. Prominence is given to the computation of a multitude of Jacobians; determinantal point processes and orthogonal polynomials of one variable; the Selberg integral, Jack polynomials, and generalized hypergeometric functions; Painleve transcendents; macroscopic electrostatistics and asymptotic formulas; nonintersecting paths and models in statistical mechanics; and applications of random matrix theory.This is the first textbook development of both nonsymmetric and symmetric Jack polynomial theory, as well as the connection between Selberg integral theory and beta ensembles. The author provides hundreds of guided exercises and linked topics, making Log-Gases and Random Matrices an indispensable reference work, as well as a learning resource for all students and researchers in the field.

This book gives a comprehensive and self-contained introduction to the theory of symmetric Markov processes and symmetric quasi-regular Dirichlet forms. In a detailed and accessible manner, Zhen-Qing Chen and Masatoshi Fukushima cover the essential elements and applications of the theory of symmetric Markov processes, including recurrence/transience criteria, probabilistic potential theory, additive functional theory, and time change theory. The authors develop the theory in a general framework of symmetric quasi-regular Dirichlet forms in a unified manner with that of regular Dirichlet forms, emphasizing the role of extended Dirichlet spaces and the rich interplay between the probabilistic and analytic aspects of the theory. Chen and Fukushima then address the latest advances in the theory, presented here for the first time in any book. Topics include the characterization of time-changed Markov processes in terms of Douglas integrals and a systematic account of reflected Dirichlet spaces, and the important roles such advances play in the boundary theory of symmetric Markov processes.This volume is an ideal resource for researchers and practitioners, and can also serve as a textbook for advanced graduate students. It includes examples, appendixes, and exercises with solutions.

By studying the degeneration of abelian varieties with PEL structures, this book explains the compactifications of smooth integral models of all PEL-type Shimura varieties, providing the logical foundation for several exciting recent developments. The book is designed to be accessible to graduate students who have an understanding of schemes and abelian varieties. PEL-type Shimura varieties, which are natural generalizations of modular curves, are useful for studying the arithmetic properties of automorphic forms and automorphic representations, and they have played important roles in the development of the Langlands program. As with modular curves, it is desirable to have integral models of compactifications of PEL-type Shimura varieties that can be described in sufficient detail near the boundary.This book explains in detail the following topics about PEL-type Shimura varieties and their compactifications: * A construction of smooth integral models of PEL-type Shimura varieties by defining and representing moduli problems of abelian schemes with PEL structures * An analysis of the degeneration of abelian varieties with PEL structures into semiabelian schemes, over noetherian normal complete adic base rings * A construction of toroidal and minimal compactifications of smooth integral models of PEL-type Shimura varieties, with detailed descriptions of their structure near the boundary Through these topics, the book generalizes the theory of degenerations of polarized abelian varieties and the application of that theory to the construction of toroidal and minimal compactifications of Siegel moduli schemes over the integers (as developed by Mumford, Faltings, and Chai).

This book seeks to describe the rapid development in recent decades of sieve methods able to detect prime numbers. The subject began with Eratosthenes in antiquity, took on new shape with Legendre's form of the sieve, was substantially reworked by Ivan M. Vinogradov and Yuri V. Linnik, but came into its own with Robert C. Vaughan and important contributions from others, notably Roger Heath-Brown and Henryk Iwaniec. Prime-Detecting Sieves breaks new ground by bringing together several different types of problems that have been tackled with modern sieve methods and by discussing the ideas common to each, in particular the use of Type I and Type II information.No other book has undertaken such a systematic treatment of prime-detecting sieves. Among the many topics Glyn Harman covers are primes in short intervals, the greatest prime factor of the sequence of shifted primes, Goldbach numbers in short intervals, the distribution of Gaussian primes, and the recent work of John Friedlander and Iwaniec on primes that are a sum of a square and a fourth power, and Heath-Brown's work on primes represented as a cube plus twice a cube. This book contains much that is accessible to beginning graduate students, yet also provides insights that will benefit established researchers.