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| banach algebra book: A Course in Commutative Banach Algebras Eberhard Kaniuth, 2008-12-16 Banach algebras are Banach spaces equipped with a continuous multipli- tion. In roughterms,there arethree types ofthem:algebrasofboundedlinear operators on Banach spaces with composition and the operator norm, al- bras consisting of bounded continuous functions on topological spaces with pointwise product and the uniform norm, and algebrasof integrable functions on locally compact groups with convolution as multiplication. These all play a key role in modern analysis. Much of operator theory is best approached from a Banach algebra point of view and many questions in complex analysis (such as approximation by polynomials or rational functions in speci?c - mains) are best understood within the framework of Banach algebras. Also, the study of a locally compact Abelian group is closely related to the study 1 of the group algebra L (G). There exist a rich literature and excellent texts on each single class of Banach algebras, notably on uniform algebras and on operator algebras. This work is intended as a textbook which provides a thorough introduction to the theory of commutative Banach algebras and stresses the applications to commutative harmonic analysis while also touching on uniform algebras. In this sense and purpose the book resembles Larsen’s classical text [75] which shares many themes and has been a valuable resource. However, for advanced graduate students and researchers I have covered several topics which have not been published in books before, including some journal articles. |
| banach algebra book: Banach Algebras and the General Theory of *-Algebras: Volume 1, Algebras and Banach Algebras Theodore W. Palmer, 1994-03-25 This is the first volume of a two volume set that provides a modern account of basic Banach algebra theory including all known results on general Banach *-algebras. This account emphasizes the role of *-algebraic structure and explores the algebraic results that underlie the theory of Banach algebras and *-algebras. The first volume, which contains previously unpublished results, is an independent, self-contained reference on Banach algebra theory. Each topic is treated in the maximum interesting generality within the framework of some class of complex algebras rather than topological algebras. Proofs are presented in complete detail at a level accessible to graduate students. The book contains a wealth of historical comments, background material, examples, particularly in noncommutative harmonic analysis, and an extensive bibliography. Volume II is forthcoming. |
| banach algebra book: Complete Normed Algebras Frank F. Bonsall, John Duncan, 2012-12-06 The axioms of a complex Banach algebra were very happily chosen. They are simple enough to allow wide ranging fields of application, notably in harmonic analysis, operator theory and function algebras. At the same time they are tight enough to allow the development of a rich collection of results, mainly through the interplay of the elementary parts of the theories of analytic functions, rings, and Banach spaces. Many of the theorems are things of great beauty, simple in statement, surprising in content, and elegant in proof. We believe that some of them deserve to be known by every mathematician. The aim of this book is to give an account of the principal methods and results in the theory of Banach algebras, both commutative and non commutative. It has been necessary to apply certain exclusion principles in order to keep our task within bounds. Certain classes of concrete Banach algebras have a very rich literature, namely C*-algebras, function algebras, and group algebras. We have regarded these highly developed theories as falling outside our scope. We have not entirely avoided them, but have been concerned with their place in the general theory, and have stopped short of developing their special properties. For reasons of space and time we have omitted certain other topics which would quite naturally have been included, in particular the theories of multipliers and of extensions of Banach algebras, and the implications for Banach algebras of some of the standard algebraic conditions on rings. |
| banach algebra book: Banach Algebras Wiesław Żelazko, 1973 |
| banach algebra book: General Theory of Banach Algebras Charles Earl Rickart, 1974 |
| banach algebra book: Banach Algebras and the General Theory of *-Algebras: Volume 2, *-Algebras Theodore W. Palmer, 1994 This is the second volume of a two-volume set that provides a modern account of basic Banach algebra theory including all known results on general Banach *-algebras. The author emphasizes the roles of *-algebra structure and explores the algebraic results which underlie the theory of Banach algebras and *-algebras. Proofs are presented in complete detail at a level accessible to graduate students. The books will become the standard reference for the general theory of *-algebras. This second volume deals with *-algebras. Chapter 9 develops the theory of *-algebras without additional restrictions. Chapter 10 proves nearly all the results previously known for Banach *-algebras and hermitian Banach *-algebras for *-algebras with various essentially algebraic restrictions. Chapter 11 restates the previous results in terms of Banach *-algebras and uses them to prove results explicitly involving the complete norm. Chapter 12 is devoted to locally compact groups and the *-algebras related to them. |
| banach algebra book: Banach Algebras and Several Complex Variables John Wermer, 2013-06-29 During the past twenty years many connections have been found between the theory of analytic functions of one or more complex variables and the study of commutative Banach algebras. On the one hand, function theory has been used to answer algebraic questions such as the question of the existence of idempotents in a Banach algebra. On the other hand, concepts arising from the study of Banach algebras such as the maximal ideal space, the Silov boundary, Gleason parts, etc. have led to new questions and to new methods of proof in function theory. Roughly one third of this book isconcerned with developing some of the principal applications of function theory in several complex variables to Banach algebras. We presuppose no knowledge of severalcomplex variables on the part of the reader but develop the necessary material from scratch. The remainder of the book deals with problems of uniform approximation on compact subsets of the space of n complex variables. For n > I no complete theory exists but many important particular problems have been solved. Throughout, our aim has been to make the exposition elementary and self-contained. We have cheerfully sacrificed generality and completeness all along the way in order to make it easier to understand the main ideas. |
| banach algebra book: Banach algebra techniques in operator theory R. G. Douglas, 1976 |
| banach algebra book: Introduction to Banach Algebras, Operators, and Harmonic Analysis H. Garth Dales, 2003-11-13 This work has arisen from lecture courses given by the authors on important topics within functional analysis. The authors, who are all leading researchers, give introductions to their subjects at a level ideal for beginning graduate students, and others interested in the subject. The collection has been carefully edited so as to form a coherent and accessible introduction to current research topics. The first chapter by Professor Dales introduces the general theory of Banach algebras, which serves as a background to the remaining material. Dr Willis then studies a centrally important Banach algebra, the group algebra of a locally compact group. The remaining chapters are devoted to Banach algebras of operators on Banach spaces: Professor Eschmeier gives all the background for the exciting topic of invariant subspaces of operators, and discusses some key open problems; Dr Laursen and Professor Aiena discuss local spectral theory for operators, leading into Fredholm theory. |
| banach algebra book: Banach Algebras and Applications Mahmoud Filali, 2020-08-24 Banach algebras is a multilayered area in mathematics with many ramifications. With a diverse coverage of different schools working on the subject, this proceedings volume reflects recent achievements in areas such as Banach algebras over groups, abstract harmonic analysis, group actions, amenability, topological homology, Arens irregularity, C*-algebras and dynamical systems, operator theory, operator spaces, and locally compact quantum groups. |
| banach algebra book: A Short Course on Spectral Theory William Arveson, 2001-11-09 This book presents the basic tools of modern analysis within the context of the fundamental problem of operator theory: to calculate spectra of specific operators on infinite dimensional spaces, especially operators on Hilbert spaces. The tools are diverse, and they provide the basis for more refined methods that allow one to approach problems that go well beyond the computation of spectra: the mathematical foundations of quantum physics, noncommutative K-theory, and the classification of simple C*-algebras being three areas of current research activity which require mastery of the material presented here. |
| banach algebra book: Homogeneous Banach Algebras Hwai-Chiuan Wang, 2020-11-25 This book examines some aspects of homogeneous Banach algebras and related topics to illustrate various methods used in several classes of group algebras. It guides the reader toward some of the problems in harmonic analysis such as the problems of factorizations and closed subalgebras. |
| banach algebra book: Introduction to Banach Spaces and Algebras Graham Allan, 2010-11-04 Banach spaces and algebras are a key topic of pure mathematics. Graham Allan's careful and detailed introductory account will prove essential reading for anyone wishing to specialise in functional analysis and is aimed at final year undergraduates or masters level students. Based on the author's lectures to fourth year students at Cambridge University, the book assumes knowledge typical of first degrees in mathematics, including metric spaces, analytic topology, and complex analysis. However, readers are not expected to be familiar with the Lebesgue theory of measure and integration. The text begins by giving the basic theory of Banach spaces, including dual spaces and bounded linear operators. It establishes forms of the theorems that are the pillars of functional analysis, including the Banach-Alaoglu, Hahn-Banach, uniform boundedness, open mapping, and closed graph theorems. There are applications to Fourier series and operators on Hilbert spaces. The main body of the text is an introduction to the theory of Banach algebras. A particular feature is the detailed account of the holomorphic functional calculus in one and several variables; all necessary background theory in one and several complex variables is fully explained, with many examples and applications considered. Throughout, exercises at sections ends help readers test their understanding, while extensive notes point to more advanced topics and sources. The book was edited for publication by Professor H. G. Dales of Leeds University, following the death of the author in August, 2007. |
| banach algebra book: Banach Algebras Ronald Larsen, 1973 |
| banach algebra book: Banach Algebras and Their Applications Anthony To-Ming Lau, Volker Runde, 2004 This proceedings volume is from the international conference on Banach Algebras and Their Applications held at the University of Alberta (Edmonton). It contains a collection of refereed research papers and high-level expository articles that offer a panorama of Banach algebra theory and its manifold applications. Topics in the book range from - theory to abstract harmonic analysis to operator theory. It is suitable for graduate students and researchers interested in Banach algebras. |
| banach algebra book: Amenable Banach Algebras Volker Runde, 2020-03-03 This volume provides readers with a detailed introduction to the amenability of Banach algebras and locally compact groups. By encompassing important foundational material, contemporary research, and recent advancements, this monograph offers a state-of-the-art reference. It will appeal to anyone interested in questions of amenability, including those familiar with the author’s previous volume Lectures on Amenability. Cornerstone topics are covered first: namely, the theory of amenability, its historical context, and key properties of amenable groups. This introduction leads to the amenability of Banach algebras, which is the main focus of the book. Dual Banach algebras are given an in-depth exploration, as are Banach spaces, Banach homological algebra, and more. By covering amenability’s many applications, the author offers a simultaneously expansive and detailed treatment. Additionally, there are numerous exercises and notes at the end of every chapter that further elaborate on the chapter’s contents. Because it covers both the basics and cutting edge research, Amenable Banach Algebras will be indispensable to both graduate students and researchers working in functional analysis, harmonic analysis, topological groups, and Banach algebras. Instructors seeking to design an advanced course around this subject will appreciate the student-friendly elements; a prerequisite of functional analysis, abstract harmonic analysis, and Banach algebra theory is assumed. |
| banach algebra book: Banach Algebra Techniques in Operator Theory Ronald G. Douglas, 2012-12-06 Operator theory is a diverse area of mathematics which derives its impetus and motivation from several sources. It began with the study of integral equations and now includes the study of operators and collections of operators arising in various branches of physics and mechanics. The intention of this book is to discuss certain advanced topics in operator theory and to provide the necessary background for them assuming only the standard senior-first year graduate courses in general topology, measure theory, and algebra. At the end of each chapter there are source notes which suggest additional reading along with giving some comments on who proved what and when. In addition, following each chapter is a large number of problems of varying difficulty. This new edition will appeal to a new generation of students seeking an introduction to operator theory. |
| banach algebra book: Cohomology in Banach Algebras Barry Edward Johnson, 1972 Let [Fraktur capital]A be a Banach algebra, [Fraktur capital]X a two-sided Banach [Fraktur capital] A-module, and define the cohomology groups [italic]H[italic superscript]n([Fraktur capital]A, [Fraktur capital]X) from the complex of bounded cochains in exact analogy to the classical (algebraic) case. This article gives an introduction to several aspects of the resulting theory. |
| banach algebra book: Several Complex Variables and Banach Algebras Herbert Alexander, John Wermer, 2008-01-17 Many connections have been found between the theory of analytic functions of one or more complex variables and the study of commutative Banach algebras. While function theory has often been employed to answer algebraic questions such as the existence of idempotents in a Banach algebra, concepts arising from the study of Banach algebras including the maximal ideal space, the Silov boundary, Geason parts, etc. have led to new questions and to new methods of proofs in function theory. This book is concerned with developing some of the principal applications of function theory in several complex variables to Banach algebras. The authors do not presuppose any knowledge of several complex variables on the part of the reader and all relevant material is developed within the text. Furthermore, the book deals with problems of uniform approximation on compact subsets of the space of n complex variables. The third edition of this book contains new material on; maximum modulus algebras and subharmonicity, the hull of a smooth curve, integral kernels, perturbations of the Stone-Weierstrass Theorem, boundaries of analytic varieties, polynomial hulls of sets over the circle, areas, and the topology of hulls. The authors have also included a new chapter containing commentaries on history and recent developments and an updated and expanded reading list. |
| banach algebra book: M-Ideals in Banach Spaces and Banach Algebras Peter Harmand, Dirk Werner, Wend Werner, 2006-11-15 This book provides a comprehensive exposition of M-ideal theory, a branch ofgeometric functional analysis which deals with certain subspaces of Banach spaces arising naturally in many contexts. Starting from the basic definitions the authors discuss a number of examples of M-ideals (e.g. the closed two-sided ideals of C*-algebras) and develop their general theory. Besides, applications to problems from a variety of areas including approximation theory, harmonic analysis, C*-algebra theory and Banach space geometry are presented. The book is mainly intended as a reference volume for researchers working in one of these fields, but it also addresses students at the graduate or postgraduate level. Each of its six chapters is accompanied by a Notes-and-Remarks section which explores further ramifications of the subject and gives detailed references to the literature. An extensive bibliography is included. |
| banach algebra book: Homogeneous Banach Algebras Wang, 2020-11-25 This book examines some aspects of homogeneous Banach algebras and related topics to illustrate various methods used in several classes of group algebras. It guides the reader toward some of the problems in harmonic analysis such as the problems of factorizations and closed subalgebras. |
| banach algebra book: Banach Algebras and the General Theory of *-Algebras: Volume 1, Algebras and Banach Algebras Theodore W. Palmer, 2009-12-03 This is the first volume of a two volume set that provides a modern account of basic Banach algebra theory including all known results on general Banach *-algebras. This account emphasizes the role of *-algebraic structure and explores the algebraic results that underlie the theory of Banach algebras and *-algebras. The first volume, which contains previously unpublished results, is an independent, self-contained reference on Banach algebra theory. Each topic is treated in the maximum interesting generality within the framework of some class of complex algebras rather than topological algebras. Proofs are presented in complete detail at a level accessible to graduate students. The book contains a wealth of historical comments, background material, examples, particularly in noncommutative harmonic analysis, and an extensive bibliography. Volume II is forthcoming. |
| banach algebra book: Real Function Algebras S.H. Kulkarni, B.V. Limaye, 2020-08-27 This self-contained reference/text presents a thorough account of the theory of real function algebras. Employing the intrinsic approach, avoiding the complexification technique, and generalizing the theory of complex function algebras, this single-source volume includes: an introduction to real Banach algebras; various generalizations of the Stone-Weierstrass theorem; Gleason parts; Choquet and Shilov boundaries; isometries of real function algebras; extensive references; and a detailed bibliography.;Real Function Algebras offers results of independent interest such as: topological conditions for the commutativity of a real or complex Banach algebra; Ransford's short elementary proof of the Bishop-Stone-Weierstrass theorem; the implication of the analyticity or antianalyticity of f from the harmonicity of Re f, Re f(2), Re f(3), and Re f(4); and the positivity of the real part of a linear functional on a subspace of C(X).;With over 600 display equations, this reference is for mathematical analysts; pure, applied, and industrial mathematicians; and theoretical physicists; and a text for courses in Banach algebras and function algebras. |
| banach algebra book: Banach Algebras and the General Theory of *-Algebras: Volume 2, *-Algebras Theodore W. Palmer, 2001-04-23 This second volume of a two-volume set provides a modern account of basic Banach algebra theory including all known results on general Banach *-algebras. The author emphasizes the roles of *-algebra structure and explores the algebraic results that underlie the theory of Banach algebras and *-algebras. Proofs are presented in complete detail at a level accessible to graduate students. The books contain a wealth of historical comments, background material, examples, and an extensive bibliography. Together they constitute the standard reference for the general theory of *-algebras. This second volume deals with *-algebras. Noteworthy chapters develop the theory of *-algebras without additional restrictions, going well beyond what has been proved previously in this context and describe locally compact groups and the *-algebras related to them. |
| banach algebra book: Functional Analysis Joseph Muscat, 2024-02-28 This textbook provides an introduction to functional analysis suitable for lecture courses to final year undergraduates or beginning graduates. Starting from the very basics of metric spaces, the book adopts a self-contained approach to Banach spaces and operator theory that covers the main topics, including the spectral theorem, the Gelfand transform, and Banach algebras. Various applications, such as least squares approximation, inverse problems, and Tikhonov regularization, illustrate the theory. Over 1000 worked examples and exercises of varying difficulty present the reader with ample material for reflection. This new edition of Functional Analysis has been completely revised and corrected, with many passages rewritten for clarity, numerous arguments simplified, and a good amount of new material added, including new examples and exercises. The prerequisites, however, remain the same with only knowledge of linear algebra and real analysis of a singlevariable assumed of the reader. |
| banach algebra book: Banach Algebras with Symbol and Singular Integral Operators N. Krupnik, 2013-11-22 About fifty years aga S. G. Mikhlin, in solving the regularization problem for two-dimensional singular integral operators [56], assigned to each such operator a func tion which he called a symbol, and showed that regularization is possible if the infimum of the modulus of the symbol is positive. Later, the notion of a symbol was extended to multidimensional singular integral operators (of arbitrary dimension) [57, 58, 21, 22]. Subsequently, the synthesis of singular integral, and differential operators [2, 8, 9]led to the theory of pseudodifferential operators [17, 35] (see also [35(1)-35(17)]*), which are naturally characterized by their symbols. An important role in the construction of symbols for many classes of operators was played by Gelfand's theory of maximal ideals of Banach algebras [201. Using this the ory, criteria were obtained for Fredholmness of one-dimensional singular integral operators with continuous coefficients [34 (42)], Wiener-Hopf operators [37], and multidimensional singular integral operators [38 (2)]. The investigation of systems of equations involving such operators has led to the notion of matrix symbol [59, 12 (14), 39, 41]. This notion plays an essential role not only for systems, but also for singular integral operators with piecewise-continuous (scalar) coefficients [44 (4)]. At the same time, attempts to introduce a (scalar or matrix) symbol for other algebras have failed. |
| banach algebra book: Banach Algebras on Semigroups and on Their Compactifications Harold G. Dales, Anthony To-Ming Lau, Dona Strauss, 2010 Volume 205, number 966 (end of volume). |
| banach algebra book: Proceedings of the Conference on Banach Algebras and Several Complex Variables Frederick P. Greenleaf, Denny Gulick, 1984 Contains papers presented at the conference on Banach Algebras and Several Complex Variables held June 21-24, 1983, to honor Professor Charles E Rickart upon his retirement from Yale University. This work includes articles that present advances in topics related to Banach algebras, function algebras and infinite dimensional holomorphy. |
| banach algebra book: Algebraic and Strong Splittings of Extensions of Banach Algebras William G. Bade, Harold G. Dales, Zinaida Alexandrovna Lykova, 1999 In this volume, the authors address the following: Let $A$ be a Banach algebra, and let $\sum\:\ 0\rightarrow I\rightarrow\frak A\overset\pi\to\longrightarrow A\rightarrow 0$ be an extension of $A$, where $\frak A$ is a Banach algebra and $I$ is a closed ideal in $\frak A$. The extension splits algebraically (respectively, splits strongly) if there is a homomorphism (respectively, continuous homomorphism) $\theta\: A\rightarrow\frak A$ such that $\pi\circ\theta$ is the identity on $A$. Consider first for which Banach algebras $A$ it is true that every extension of $A$ in a particular class of extensions splits, either algebraically or strongly, and second for which Banach algebras it is true that every extension of $A$ in a particular class which splits algebraically also splits strongly. These questions are closely related to the question when the algebra $\frak A$ has a (strong) Wedderburn decomposition. The main technique for resolving these questions involves the Banach cohomology group $\cal H2(A,E)$ for a Banach $A$-bimodule $E$, and related cohomology groups. Later chapters are particularly concerned with the case where the ideal $I$ is finite-dimensional. Results are obtained for many of the standard Banach algebras $A$. |
| banach algebra book: Banach Algebra Techniques in the Theory of Toeplitz Operators Ronald G. Douglas, 1980 |
| banach algebra book: Perturbation of Banach Algebras Krzysztof Jarosz, 2006-11-14 |
| banach algebra book: Banach Algebras and Applications Mahmoud Filali, 2020-08-24 Banach algebras is a multilayered area in mathematics with many ramifications. With a diverse coverage of different schools working on the subject, this proceedings volume reflects recent achievements in areas such as Banach algebras over groups, abstract harmonic analysis, group actions, amenability, topological homology, Arens irregularity, C*-algebras and dynamical systems, operator theory, operator spaces, and locally compact quantum groups. |
| banach algebra book: Calkin Algebras and Algebras of Operators on Banach SPates Caradus, 1974-09-01 Since the appearance of Banach algebra theory, the interaction between the theories ofBanach algebras with involution and that of bounded linear operators on a Hilbert space hasbeen extensively developed. The connections of Banach algebras with the theory ofbounded linear operators on a Hilbert space have also evolved, and Calkin Algebras andAlgebras of Operators on Banach Spaces provides an introduction to this set of ideas.The book begins with a treatment of the classical Riesz-Schauder theory which takesadvantage of the most recent developments-some of this material appears here for the firsttime. Although the reader should be familiar with the basics of functional analysis, anintroductory chapter on Banach algebras has been included. Other topics dealt with includeFredholm operators, semi-Fredholm operators, Riesz operators. and Calkin algebras.This volume will be of direct interest to both graduate students and research mathematicians. |
| banach algebra book: Derivations and Automorphisms of Banach Algebras of Power Series Sandy Grabiner, 1974 This paper studies derivations, endomorphisms, automorphisms, and various related questions about certain Banach algebras, B, which are continuously embedded in the space of complex formal power series in the indeterminate z. |
| banach algebra book: Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras Vladimir Müller, 2013-11-11 Spectral theory is an important part of functional analysis.It has numerous appli cations in many parts of mathematics and physics including matrix theory, func tion theory, complex analysis, differential and integral equations, control theory and quantum physics. In recent years, spectral theory has witnessed an explosive development. There are many types of spectra, both for one or several commuting operators, with important applications, for example the approximate point spectrum, Taylor spectrum, local spectrum, essential spectrum, etc. The present monograph is an attempt to organize the available material most of which exists only in the form of research papers scattered throughout the literature. The aim is to present a survey of results concerning various types of spectra in a unified, axiomatic way. The central unifying notion is that of a regularity, which in a Banach algebra is a subset of elements that are considered to be nice. A regularity R in a Banach algebra A defines the corresponding spectrum aR(a) = {A E C : a - ,\ rJ. R} in the same way as the ordinary spectrum is defined by means of invertible elements, a(a) = {A E C : a - ,\ rJ. Inv(A)}. Axioms of a regularity are chosen in such a way that there are many natural interesting classes satisfying them. At the same time they are strong enough for non-trivial consequences, for example the spectral mapping theorem. |
| banach algebra book: Modular Representation Theory and Commutative Banach Algebras David J. Benson, 2024-07-25 View the abstract. |
| banach algebra book: M-ideals in Banach Spaces and Banach Algebras Peter Harmand, Dirk Werner, Wend Werner, 1993 |
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