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K Theory And C Algebras
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Book Synopsis An Introduction to K-Theory for C*-Algebras by : M. Rørdam
Download or read book An Introduction to K-Theory for C*-Algebras written by M. Rørdam and published by Cambridge University Press. This book was released on 2000-07-20 with total page 260 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a very elementary introduction to K-theory for C*-algebras, and is ideal for beginning graduate students.
Author :Niels Erik Wegge-Olsen Publisher :Oxford University Press on Demand ISBN 13 :9780198596943 Total Pages :370 pages Book Rating :4.5/5 (969 download)
Book Synopsis K-theory and C*-algebras by : Niels Erik Wegge-Olsen
Download or read book K-theory and C*-algebras written by Niels Erik Wegge-Olsen and published by Oxford University Press on Demand. This book was released on 1993 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: K-theory is often considered a complicated mathematical theory for specialists only. This book is an accessible introduction to the basics and provides detailed explanations of the various concepts required for a deeper understanding of the subject. Some familiarity with basic C*algebra theory is assumed. The book then follows a careful construction and analysis of the operator K-theory groups and proof of the results of K-theory, including Bott periodicity. Of specific interest to algebraists and geometrists, the book aims to give full instruction. No details are left out in the presentation and many instructive and generously hinted exercises are provided. Apart from K-theory, this book offers complete and self contained expositions of important advanced C*-algebraic constructions like tensor products, multiplier algebras and Hilbert modules.
Book Synopsis K-Theory for Operator Algebras by : Bruce Blackadar
Download or read book K-Theory for Operator Algebras written by Bruce Blackadar and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 347 pages. Available in PDF, EPUB and Kindle. Book excerpt: K -Theory has revolutionized the study of operator algebras in the last few years. As the primary component of the subject of "noncommutative topol ogy," K -theory has opened vast new vistas within the structure theory of C* algebras, as well as leading to profound and unexpected applications of opera tor algebras to problems in geometry and topology. As a result, many topolo gists and operator algebraists have feverishly begun trying to learn each others' subjects, and it appears certain that these two branches of mathematics have become deeply and permanently intertwined. Despite the fact that the whole subject is only about a decade old, operator K -theory has now reached a state of relative stability. While there will undoubtedly be many more revolutionary developments and applications in the future, it appears the basic theory has more or less reached a "final form." But because of the newness of the theory, there has so far been no comprehensive treatment of the subject. It is the ambitious goal of these notes to fill this gap. We will develop the K -theory of Banach algebras, the theory of extensions of C*-algebras, and the operator K -theory of Kasparov from scratch to its most advanced aspects. We will not treat applications in detail; however, we will outline the most striking of the applications to date in a section at the end, as well as mentioning others at suitable points in the text.
Book Synopsis C*-Algebras and Operator Theory by : Gerald J. Murphy
Download or read book C*-Algebras and Operator Theory written by Gerald J. Murphy and published by Academic Press. This book was released on 2014-06-28 with total page 297 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes a first- or second-year graduate course in operator theory. It is a field that has great importance for other areas of mathematics and physics, such as algebraic topology, differential geometry, and quantum mechanics. It assumes a basic knowledge in functional analysis but no prior acquaintance with operator theory is required.
Book Synopsis Noncommutative Geometry by : Alain Connes
Download or read book Noncommutative Geometry written by Alain Connes and published by Springer. This book was released on 2003-12-15 with total page 364 pages. Available in PDF, EPUB and Kindle. Book excerpt: Noncommutative Geometry is one of the most deep and vital research subjects of present-day Mathematics. Its development, mainly due to Alain Connes, is providing an increasing number of applications and deeper insights for instance in Foliations, K-Theory, Index Theory, Number Theory but also in Quantum Physics of elementary particles. The purpose of the Summer School in Martina Franca was to offer a fresh invitation to the subject and closely related topics; the contributions in this volume include the four main lectures, cover advanced developments and are delivered by prominent specialists.
Author :Kenneth R. Davidson Publisher :American Mathematical Society, Fields Institute ISBN 13 :1470475081 Total Pages :325 pages Book Rating :4.4/5 (74 download)
Book Synopsis C*-Algebras by Example by : Kenneth R. Davidson
Download or read book C*-Algebras by Example written by Kenneth R. Davidson and published by American Mathematical Society, Fields Institute. This book was released on 2023-10-04 with total page 325 pages. Available in PDF, EPUB and Kindle. Book excerpt: The subject of C*-algebras received a dramatic revitalization in the 1970s by the introduction of topological methods through the work of Brown, Douglas, and Fillmore on extensions of C*-algebras and Elliott's use of $K$-theory to provide a useful classification of AF algebras. These results were the beginning of a marvelous new set of tools for analyzing concrete C*-algebras. This book is an introductory graduate level text which presents the basics of the subject through a detailed analysis of several important classes of C*-algebras. The development of operator algebras in the last twenty years has been based on a careful study of these special classes. While there are many books on C*-algebras and operator algebras available, this is the first one to attempt to explain the real examples that researchers use to test their hypotheses. Topics include AF algebras, Bunce–Deddens and Cuntz algebras, the Toeplitz algebra, irrational rotation algebras, group C*-algebras, discrete crossed products, abelian C*-algebras (spectral theory and approximate unitary equivalence) and extensions. It also introduces many modern concepts and results in the subject such as real rank zero algebras, topological stable rank, quasidiagonality, and various new constructions. These notes were compiled during the author's participation in the special year on C*-algebras at The Fields Institute for Research in Mathematical Sciences during the 1994–1995 academic year. The field of C*-algebras touches upon many other areas of mathematics such as group representations, dynamical systems, physics, $K$-theory, and topology. The variety of examples offered in this text expose the student to many of these connections. Graduate students with a solid course in functional analysis should be able to read this book. This should prepare them to read much of the current literature. This book is reasonably self-contained, and the author has provided results from other areas when necessary.
Book Synopsis Operator Algebras by : Bruce Blackadar
Download or read book Operator Algebras written by Bruce Blackadar and published by Taylor & Francis. This book was released on 2006 with total page 552 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers a comprehensive introduction to the general theory of C*-algebras and von Neumann algebras. Beginning with the basics, the theory is developed through such topics as tensor products, nuclearity and exactness, crossed products, K-theory, and quasidiagonality. The presentation carefully and precisely explains the main features of each part of the theory of operator algebras; most important arguments are at least outlined and many are presented in full detail.
Download or read book The $K$-book written by Charles A. Weibel and published by American Mathematical Soc.. This book was released on 2013-06-13 with total page 634 pages. Available in PDF, EPUB and Kindle. Book excerpt: Informally, $K$-theory is a tool for probing the structure of a mathematical object such as a ring or a topological space in terms of suitably parameterized vector spaces and producing important intrinsic invariants which are useful in the study of algebr
Book Synopsis An Introduction to C*-Algebras and the Classification Program by : Karen R. Strung
Download or read book An Introduction to C*-Algebras and the Classification Program written by Karen R. Strung and published by Springer Nature. This book was released on 2020-12-15 with total page 322 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is directed towards graduate students that wish to start from the basic theory of C*-algebras and advance to an overview of some of the most spectacular results concerning the structure of nuclear C*-algebras. The text is divided into three parts. First, elementary notions, classical theorems and constructions are developed. Then, essential examples in the theory, such as crossed products and the class of quasidiagonal C*-algebras, are examined, and finally, the Elliott invariant, the Cuntz semigroup, and the Jiang-Su algebra are defined. It is shown how these objects have played a fundamental role in understanding the fine structure of nuclear C*-algebras. To help understanding the theory, plenty of examples, treated in detail, are included. This volume will also be valuable to researchers in the area as a reference guide. It contains an extensive reference list to guide readers that wish to travel further.
Book Synopsis K-Theory for Group C*-Algebras and Semigroup C*-Algebras by : Joachim Cuntz
Download or read book K-Theory for Group C*-Algebras and Semigroup C*-Algebras written by Joachim Cuntz and published by Birkhäuser. This book was released on 2017-10-24 with total page 325 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives an account of the necessary background for group algebras and crossed products for actions of a group or a semigroup on a space and reports on some very recently developed techniques with applications to particular examples. Much of the material is available here for the first time in book form. The topics discussed are among the most classical and intensely studied C*-algebras. They are important for applications in fields as diverse as the theory of unitary group representations, index theory, the topology of manifolds or ergodic theory of group actions. Part of the most basic structural information for such a C*-algebra is contained in its K-theory. The determination of the K-groups of C*-algebras constructed from group or semigroup actions is a particularly challenging problem. Paul Baum and Alain Connes proposed a formula for the K-theory of the reduced crossed product for a group action that would permit, in principle, its computation. By work of many hands, the formula has by now been verified for very large classes of groups and this work has led to the development of a host of new techniques. An important ingredient is Kasparov's bivariant K-theory. More recently, also the C*-algebras generated by the regular representation of a semigroup as well as the crossed products for actions of semigroups by endomorphisms have been studied in more detail. Intriguing examples of actions of such semigroups come from ergodic theory as well as from algebraic number theory. The computation of the K-theory of the corresponding crossed products needs new techniques. In cases of interest the K-theory of the algebras reflects ergodic theoretic or number theoretic properties of the action.
Download or read book K-theory written by Michael Atiyah and published by CRC Press. This book was released on 2018-03-05 with total page 181 pages. Available in PDF, EPUB and Kindle. Book excerpt: These notes are based on the course of lectures I gave at Harvard in the fall of 1964. They constitute a self-contained account of vector bundles and K-theory assuming only the rudiments of point-set topology and linear algebra. One of the features of the treatment is that no use is made of ordinary homology or cohomology theory. In fact, rational cohomology is defined in terms of K-theory.The theory is taken as far as the solution of the Hopf invariant problem and a start is mode on the J-homomorphism. In addition to the lecture notes proper, two papers of mine published since 1964 have been reproduced at the end. The first, dealing with operations, is a natural supplement to the material in Chapter III. It provides an alternative approach to operations which is less slick but more fundamental than the Grothendieck method of Chapter III, and it relates operations and filtration. Actually, the lectures deal with compact spaces, not cell-complexes, and so the skeleton-filtration does not figure in the notes. The second paper provides a new approach to K-theory and so fills an obvious gap in the lecture notes.
Book Synopsis An Introduction to the Classification of Amenable C*-algebras by : Huaxin Lin
Download or read book An Introduction to the Classification of Amenable C*-algebras written by Huaxin Lin and published by World Scientific. This book was released on 2001 with total page 336 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory and applications of C Oeu -algebras are related to fields ranging from operator theory, group representations and quantum mechanics, to non-commutative geometry and dynamical systems. By Gelfand transformation, the theory of C Oeu -algebras is also regarded as non-commutative topology. About a decade ago, George A. Elliott initiated the program of classification of C Oeu -algebras (up to isomorphism) by their K -theoretical data. It started with the classification of AT -algebras with real rank zero. Since then great efforts have been made to classify amenable C Oeu -algebras, a class of C Oeu -algebras that arises most naturally. For example, a large class of simple amenable C Oeu -algebras is discovered to be classifiable. The application of these results to dynamical systems has been established. This book introduces the recent development of the theory of the classification of amenable C Oeu -algebras OCo the first such attempt. The first three chapters present the basics of the theory of C Oeu -algebras which are particularly important to the theory of the classification of amenable C Oeu -algebras. Chapter 4 otters the classification of the so-called AT -algebras of real rank zero. The first four chapters are self-contained, and can serve as a text for a graduate course on C Oeu -algebras. The last two chapters contain more advanced material. In particular, they deal with the classification theorem for simple AH -algebras with real rank zero, the work of Elliott and Gong. The book contains many new proofs and some original results related to the classification of amenable C Oeu -algebras. Besides being as an introduction to the theory of the classification of amenable C Oeu -algebras, it is a comprehensive reference for those more familiar with the subject. Sample Chapter(s). Chapter 1.1: Banach algebras (260 KB). Chapter 1.2: C*-algebras (210 KB). Chapter 1.3: Commutative C*-algebras (212 KB). Chapter 1.4: Positive cones (207 KB). Chapter 1.5: Approximate identities, hereditary C*-subalgebras and quotients (230 KB). Chapter 1.6: Positive linear functionals and a Gelfand-Naimark theorem (235 KB). Chapter 1.7: Von Neumann algebras (234 KB). Chapter 1.8: Enveloping von Neumann algebras and the spectral theorem (217 KB). Chapter 1.9: Examples of C*-algebras (270 KB). Chapter 1.10: Inductive limits of C*-algebras (252 KB). Chapter 1.11: Exercises (220 KB). Chapter 1.12: Addenda (168 KB). Contents: The Basics of C Oeu -Algebras; Amenable C Oeu -Algebras and K -Theory; AF- Algebras and Ranks of C Oeu -Algebras; Classification of Simple AT -Algebras; C Oeu -Algebra Extensions; Classification of Simple Amenable C Oeu -Algebras. Readership: Researchers and graduate students in operator algebras."
Book Synopsis An Algebraic Introduction to K-Theory by : Bruce A. Magurn
Download or read book An Algebraic Introduction to K-Theory written by Bruce A. Magurn and published by Cambridge University Press. This book was released on 2002-05-20 with total page 704 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is an introduction to algebraic K-theory with no prerequisite beyond a first semester of algebra (including Galois theory and modules over a principal ideal domain). The presentation is almost entirely self-contained, and is divided into short sections with exercises to reinforce the ideas and suggest further lines of inquiry. No experience with analysis, geometry, number theory or topology is assumed. Within the context of linear algebra, K-theory organises and clarifies the relations among ideal class groups, group representations, quadratic forms, dimensions of a ring, determinants, quadratic reciprocity and Brauer groups of fields. By including introductions to standard algebra topics (tensor products, localisation, Jacobson radical, chain conditions, Dedekind domains, semi-simple rings, exterior algebras), the author makes algebraic K-theory accessible to first-year graduate students and other mathematically sophisticated readers. Even if your algebra is rusty, you can read this book; the necessary background is here, with proofs.
Book Synopsis C*-algebra Extensions and K-homology by : Ronald G. Douglas
Download or read book C*-algebra Extensions and K-homology written by Ronald G. Douglas and published by Princeton University Press. This book was released on 1980-07-21 with total page 112 pages. Available in PDF, EPUB and Kindle. Book excerpt: Recent developments in diverse areas of mathematics suggest the study of a certain class of extensions of C*-algebras. Here, Ronald Douglas uses methods from homological algebra to study this collection of extensions. He first shows that equivalence classes of the extensions of the compact metrizable space X form an abelian group Ext (X). Second, he shows that the correspondence X ⃗ Ext (X) defines a homotopy invariant covariant functor which can then be used to define a generalized homology theory. Establishing the periodicity of order two, the author shows, following Atiyah, that a concrete realization of K-homology is obtained.
Book Synopsis Introduction to Infinity-Categories by : Markus Land
Download or read book Introduction to Infinity-Categories written by Markus Land and published by Springer Nature. This book was released on 2021-04-21 with total page 300 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook is an introduction to the theory of infinity-categories, a tool used in many aspects of modern pure mathematics. It treats the basics of the theory and supplies all the necessary details while leading the reader along a streamlined path from the basic definitions to more advanced results such as the very important adjoint functor theorems. The book is based on lectures given by the author on the topic. While the material itself is well-known to experts, the presentation of the material is, in parts, novel and accessible to non-experts. Exercises complement this textbook that can be used both in a classroom setting at the graduate level and as an introductory text for the interested reader.
Download or read book K-Theory written by Max Karoubi and published by Springer Science & Business Media. This book was released on 2009-11-27 with total page 337 pages. Available in PDF, EPUB and Kindle. Book excerpt: From the Preface: K-theory was introduced by A. Grothendieck in his formulation of the Riemann- Roch theorem. For each projective algebraic variety, Grothendieck constructed a group from the category of coherent algebraic sheaves, and showed that it had many nice properties. Atiyah and Hirzebruch considered a topological analog defined for any compact space X, a group K{X) constructed from the category of vector bundles on X. It is this ''topological K-theory" that this book will study. Topological K-theory has become an important tool in topology. Using K- theory, Adams and Atiyah were able to give a simple proof that the only spheres which can be provided with H-space structures are S1, S3 and S7. Moreover, it is possible to derive a substantial part of stable homotopy theory from K-theory. The purpose of this book is to provide advanced students and mathematicians in other fields with the fundamental material in this subject. In addition, several applications of the type described above are included. In general we have tried to make this book self-contained, beginning with elementary concepts wherever possible; however, we assume that the reader is familiar with the basic definitions of homotopy theory: homotopy classes of maps and homotopy groups.Thus this book might be regarded as a fairly self-contained introduction to a "generalized cohomology theory".
Book Synopsis C*-Algebras and W*-Algebras by : Shoichiro Sakai
Download or read book C*-Algebras and W*-Algebras written by Shoichiro Sakai and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 271 pages. Available in PDF, EPUB and Kindle. Book excerpt: From the reviews: "This book is an excellent and comprehensive survey of the theory of von Neumann algebras. It includes all the fundamental results of the subject, and is a valuable reference for both the beginner and the expert." Mathematical Reviews
