K Theory For Operator Algebras

Author : Bruce Blackadar
Publisher : Springer Science & Business Media
Page : 347 pages
File Size : 51,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461395720

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K-Theory for Operator Algebras by Bruce Blackadar Pdf

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.

Author : M. Rørdam,Flemming Larsen,N. Laustsen
Publisher : Cambridge University Press
Page : 260 pages
File Size : 50,7 Mb
Release : 2000-07-20
Category : Mathematics
ISBN : 0521789443

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An Introduction to K-Theory for C*-Algebras by M. Rørdam,Flemming Larsen,N. Laustsen Pdf

This book provides a very elementary introduction to K-theory for C*-algebras, and is ideal for beginning graduate students.

Author : Bruce Blackadar
Publisher : Springer Science & Business Media
Page : 528 pages
File Size : 50,8 Mb
Release : 2006-03-09
Category : Mathematics
ISBN : 9783540285175

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Operator Algebras by Bruce Blackadar Pdf

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.

Author : Gerald J. Murphy
Publisher : Academic Press
Page : 296 pages
File Size : 51,5 Mb
Release : 2014-06-28
Category : Mathematics
ISBN : 9780080924960

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C*-Algebras and Operator Theory by Gerald J. Murphy Pdf

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.

Author : David E. Evans,Masamichi Takesaki
Publisher : Cambridge University Press
Page : 257 pages
File Size : 49,7 Mb
Release : 1988
Category : Mathematics
ISBN : 9780521368438

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Operator Algebras and Applications: Volume 1, Structure Theory; K-theory, Geometry and Topology by David E. Evans,Masamichi Takesaki Pdf

These volumes form an authoritative statement of the current state of research in Operator Algebras. They consist of papers arising from a year-long symposium held at the University of Warwick. Contributors include many very well-known figures in the field.

Author : Ronald G. Douglas,Claude Schochet
Publisher : American Mathematical Soc.
Page : 204 pages
File Size : 48,5 Mb
Release : 1982
Category : Mathematics
ISBN : 9780821850114

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Operator Algebras and $K$-Theory by Ronald G. Douglas,Claude Schochet Pdf

Author : Joachim Cuntz,Siegfried Echterhoff,Xin Li,Guoliang Yu
Publisher : Birkhäuser
Page : 322 pages
File Size : 53,7 Mb
Release : 2017-10-24
Category : Mathematics
ISBN : 9783319599151

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K-Theory for Group C*-Algebras and Semigroup C*-Algebras by Joachim Cuntz,Siegfried Echterhoff,Xin Li,Guoliang Yu Pdf

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.

Author : B.B. Morrel,I.M. Singer
Publisher : Springer
Page : 200 pages
File Size : 47,9 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540374237

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K-Theory and Operator Algebras by B.B. Morrel,I.M. Singer Pdf

Author : Niels Erik Wegge-Olsen
Publisher : Oxford University Press on Demand
Page : 370 pages
File Size : 53,6 Mb
Release : 1993
Category : Mathematics
ISBN : 0198596944

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K-theory and C*-algebras by Niels Erik Wegge-Olsen Pdf

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.

Author : Herbert Schröder
Publisher : Chapman and Hall/CRC
Page : 184 pages
File Size : 42,7 Mb
Release : 1993-08-23
Category : Mathematics
ISBN : UOM:39015029969386

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K-Theory for Real C*-Algebras and Applications by Herbert Schröder Pdf

This Research Note presents the K-theory and KK-theory for real C*-algebras and shows that these can be successfully applied to solve some topological problems which are not accessible to the tools developed in the complex setting alone.

Author : Alain Connes,Joachim Cuntz,Erik G. Guentner,Nigel Higson,Jerome Kaminker,John E. Roberts
Publisher : Springer Science & Business Media
Page : 372 pages
File Size : 43,9 Mb
Release : 2003-12-08
Category : Mathematics
ISBN : 3540203575

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Noncommutative Geometry by Alain Connes,Joachim Cuntz,Erik G. Guentner,Nigel Higson,Jerome Kaminker,John E. Roberts Pdf

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 : B. B. Morrel,I. M. Singer
Publisher : Unknown
Page : 204 pages
File Size : 46,5 Mb
Release : 2014-01-15
Category : Electronic
ISBN : 3662204703

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K-Theory and Operator Algebras by B. B. Morrel,I. M. Singer Pdf

Author : Kenneth R. Davidson
Publisher : American Mathematical Society, Fields Institute
Page : 325 pages
File Size : 45,7 Mb
Release : 2023-10-04
Category : Mathematics
ISBN : 9781470475086

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C*-Algebras by Example by Kenneth R. Davidson Pdf

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.

Author : Michael Atiyah
Publisher : CRC Press
Page : 138 pages
File Size : 44,9 Mb
Release : 2018-03-05
Category : Mathematics
ISBN : 9780429973178

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K-theory by Michael Atiyah Pdf

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.

Author : M. Rordam,E. Stormer
Publisher : Springer Science & Business Media
Page : 206 pages
File Size : 51,5 Mb
Release : 2013-04-18
Category : Mathematics
ISBN : 9783662048252

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Classification of Nuclear C*-Algebras. Entropy in Operator Algebras by M. Rordam,E. Stormer Pdf

to the Encyclopaedia Subseries on Operator Algebras and Non-Commutative Geometry The theory of von Neumann algebras was initiated in a series of papers by Murray and von Neumann in the 1930's and 1940's. A von Neumann algebra is a self-adjoint unital subalgebra M of the algebra of bounded operators of a Hilbert space which is closed in the weak operator topology. According to von Neumann's bicommutant theorem, M is closed in the weak operator topology if and only if it is equal to the commutant of its commutant. Afactor is a von Neumann algebra with trivial centre and the work of Murray and von Neumann contained a reduction of all von Neumann algebras to factors and a classification of factors into types I, II and III. C* -algebras are self-adjoint operator algebras on Hilbert space which are closed in the norm topology. Their study was begun in the work of Gelfand and Naimark who showed that such algebras can be characterized abstractly as involutive Banach algebras, satisfying an algebraic relation connecting the norm and the involution. They also obtained the fundamental result that a commutative unital C* -algebra is isomorphic to the algebra of complex valued continuous functions on a compact space - its spectrum. Since then the subject of operator algebras has evolved into a huge mathematical endeavour interacting with almost every branch of mathematics and several areas of theoretical physics.