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: Bruce Blackadar

Publisher: Springer Science & Business Media

ISBN: 9781461395720

Category: Mathematics

Page: 337

View: 417

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.

Since its inception by von Neumann 70 years ago, the theory of operator algebras has become a rapidly developing area of importance for the understanding of many areas of mathematics and theoretical physics.

Author: M. Takesaki

Publisher: Springer Science & Business Media

ISBN: 354042248X

Category: Mathematics

Page: 415

View: 898

Since its inception by von Neumann 70 years ago, the theory of operator algebras has become a rapidly developing area of importance for the understanding of many areas of mathematics and theoretical physics. Accessible to the non-specialist, this first part of a three-volume treatise provides a clear, carefully written survey that emphasizes the theory's analytical and topological aspects.

Together with Theory of Operator Algebras I and III, this book presents the theory of von Neumann algebras and non-commutative integration focusing on the group of automorphisms and the structure analysis.

Author: Masamichi Takesaki

Publisher: Springer Science & Business Media

ISBN: 354042914X

Category: Mathematics

Page: 518

View: 983

Together with Theory of Operator Algebras I and III, this book presents the theory of von Neumann algebras and non-commutative integration focusing on the group of automorphisms and the structure analysis. From the reviews: "These books can be warmly recommended to every graduate student who wants to become acquainted with this exciting branch of mathematics. Furthermore, they should be on the bookshelf of every researcher of the area." --ACTA SCIENTIARUM MATHEMATICARUM

In this chapter, we give a discussion of the basics of K-theory for C*-algebras, as
well as properties related to finiteness in C*-algebras, including the important
notion of quasidiagonality, which may be regarded as a type of strong finiteness.

Author: Bruce Blackadar

Publisher: Springer Science & Business Media

ISBN: 9783540285175

Category: Mathematics

Page: 528

View: 887

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.

Higher K-Theories Hyman Bass. Vol . 215 : P. Antonelli , D. ... 217 : T.J. Jech ,
Lectures in Set Theory with Particular Emphasis on the Method of Forcing . V ,
137 pages . 1971. ... 247 : Lectures on Operator Algebras . Tulane University
Ring ...

Author: Professor of Mathematics David E EvansPublish On: 1988

C*-algebra, and the comparison of the K -theory of crossed products by
homotopic actions of compact Lie groups. In some of these, particularly the last
one, it is not at all evident that inverse limits of C*-algebras should appear.
Inverse limits of ...

Author: Professor of Mathematics David E Evans

Publisher: Cambridge University Press

ISBN: 9780521368438

Category: Mathematics

Page: 256

View: 842

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.

The structure of commutator ideals is essential for investigating traces which in
turn is relevant for the calculation of the cyclic homology and the algebraic K- theory of operator ideals. Mathematics Subject Classification (2000). Primary:
47B47 ...

Author: Dumitru Gaspar

Publisher: Springer Science & Business Media

ISBN: 9783764373146

Category: Mathematics

Page: 344

View: 833

This book offers peer-reviewed articles from the 19th International Conference on Operator Theory, Summer 2002. It contains recent developments in a broad range of topics from operator theory, operator algebras and their applications, particularly to differential analysis, complex functions, ergodic theory, mathematical physics, matrix analysis, and systems theory. The book covers a large variety of topics including single operator theory, C*-algebras, diffrential operators, integral transforms, stochastic processes and operators, and more.

From this point of view, an attempt to reformulate the Baum-Connes conjecture
should involve finding a pre-C∗-norm on l1(G, A) with the property that the K- theory of the Banach algebra l1(G, A) equals the K-theory of its C∗-algebra ...

Author: Robert S. Doran

Publisher: American Mathematical Soc.

ISBN: 9781470419486

Category: Operator algebras

Page: 267

View: 712

his volume contains the proceedings of the AMS Special Session Operator Algebras and Their Applications: A Tribute to Richard V. Kadison, held from January 10–11, 2015, in San Antonio, Texas. Richard V. Kadison has been a towering figure in the study of operator algebras for more than 65 years. His research and leadership in the field have been fundamental in the development of the subject, and his influence continues to be felt though his work and the work of his many students, collaborators, and mentees. Among the topics addressed in this volume are the Kadison-Kaplanksy conjecture, classification of C∗-algebras, connections between operator spaces and parabolic induction, spectral flow, C∗-algebra actions, von Neumann algebras, and applications to mathematical physics.

JOHN VON NEUMANN AND THE THEORY OF OPERATOR ALGEBRAS D.
PETZ and M. R. RÉDEI After some earlier work ... and Murray revived much later
in connection with the Jordan operator algebras and the K - theory of C * -
algebras .

Author: John Von Neumann

Publisher: World Scientific

ISBN: 9810222017

Category: Mathematics

Page: 699

View: 532

After three decades since the first nearly complete edition of John von Neumann's papers, this book is a valuable selection of those papers and excerpts of his books that are most characteristic of his activity, and reveal that of his continuous influence.The results receiving the 1994 Nobel Prizes in economy deeply rooted in Neumann's game theory are only minor traces of his exceptionally broad spectrum of creativity and stimulation.The book is organized by the specific subjects-quantum mechanics, ergodic theory, operator algebra, hydrodynamics, economics, computers, science and society. In addition, one paper which was written in German will be translated and published in English for the first time.The sections are introduced by short explanatory notes with an emphasis on recent developments based on von Neumann's contributions. An overall picture is provided by Ulam's, one of his most intimate partners in thinking, 1958 memorial lecture. Facsimilae and translations of some of his personal letters and a newly completed bibliography based on von Neumann's own careful compilation are added.

Special Topics, Elementary Theory - An Exercise Approach Richard V. Kadison,
John R. Ringrose. 7.6.50. ... where f(j,k) = k for all j in N. From Theorem 5.6.4, Mf
is a self-adjoint operator affiliated with the multiplication algebra A of /2(N x N).

Author: Richard V. Kadison

Publisher: American Mathematical Soc.

ISBN: 0821809911

Category: Mathematics

Page: 859

View: 422

This volume is the companion volume to Fundamentals of the Theory of Operator Algebras - Volume I: Elementary Theory. The goal of the text is to teach the subject and lead readers to where the literature - in the subject specifically and in its many applications - becomes accessible. The choice of material was made from among the fundamentals of what may be called the classical theory of operator algebras. This volume contains written solutions to the exercises in the Fundamentals of the Theory of Operator Algebras. Volume I: Elementary Theory.

Using Koebe-Morse coding of geodesic lines, we establish a bijection between
Bratteli diagrams of such algebras and measured foliations. This approach
allows us to apply K-theory of operator algebras to prove a strict ergodicity
criterion and ...

Author: Joseph A. Ball

Publisher: Springer Science & Business Media

ISBN: 3764385391

Category: Mathematics

Page: 340

View: 355

This volume comprises the proceedings of the International Workshop on Operator Theory and Its Applications held at the University of Connecticut in July 2005.

In recent years, the theory of operator algebras has received a strong impetus in
Spain. This is in part due to ... These events already have a follow-up, in the form
of a school and workshop “Topics in Operator Algebras and ... Part A: K-Theory ...

Author: Ara, Pere

Publisher: American Mathematical Soc.

ISBN: 9780821849057

Category: Mathematics

Page: 168

View: 824

This volume contains survey papers on the theory of operator algebras based on lectures given at the ``Lluis Santalo'' Summer School of the Real Sociedad Matematica Espanola, held in July 2008 at the Universidad Internacional Menendez Pelayo, in Santander (Spain). Topics in this volume cover current fundamental aspects of the theory of operator algebras, which have important applications such as: $K$-Theory, the Cuntz semigroup, and Classification for $C^*$-algebras Modular Theory for von Neumann algebras and applications to Quantum Field Theory Amenability, Hyperbolic Groups, and Operator Algebras. The theory of operator algebras, introduced in the thirties by J. von Neumann and F. J. Murray, was developed in close relationship with fundamental aspects of functional analysis, ergodic theory, harmonic analysis, and quantum physics. More recently, this field has shown many other fruitful interrelations with several areas of mathematics and mathematical physics. This book is published in cooperation with Real Sociedad Matematica Espanola.

B(H 69 K) by the matrix [Tab] of the bounded operator U“1T0U acting on Eben
63H, as in the discussion preceding Proposition 2.6.13. Show that, when T0 E B(
H) ® B(K), there is a (norm-) compact subset of B(H) that contains all the entries T
...

Author: Richard V. Kadison

Publisher: American Mathematical Soc.

ISBN: 9780821894682

Category:

Page: 587

View: 536

This volume is the companion volume to Fundamentals of the Theory of Operator Algebras. Volume II--Advanced Theory (Graduate Studies in Mathematics series, Volume 16). The goal of the text proper is to teach the subject and lead readers to where the vast literature--in the subject specifically and in its many applications--becomes accessible. The choice of material was made from among the fundamentals of what may be called the "classical" theory of operator algebras. This volume contains the written solutions to the exercises in the Fundamentals of the Theory of Operator Algebras. Volume II--Advanced Theory.

These algebras can for instance arise from pseudodifferential operators,
differential forms, convolution algebras etc. ... The present note can be seen as
part of a program in which we analyze constructions, that are classical in K-theory
for ...

Author: Ola Bratteli

Publisher: Springer Science & Business Media

ISBN: 9783540341970

Category: Mathematics

Page: 279

View: 884

The theme of the first Abel Symposium was operator algebras in a wide sense. In the last 40 years operator algebras have developed from a rather special discipline within functional analysis to become a central field in mathematics often described as "non-commutative geometry". It has branched out in several sub-disciplines and made contact with other subjects. The contributions to this volume give a state-of-the-art account of some of these sub-disciplines and the variety of topics reflect to some extent how the subject has developed. This is the first volume in a prestigious new book series linked to the Abel prize.

This volume contains the proceedings of the ICM 2018 satellite school and workshop K-theory conference in Argentina.

Author: Guillermo Cortiñas

Publisher: American Mathematical Soc.

ISBN: 9781470450267

Category: Education

Page: 388

View: 966

This volume contains the proceedings of the ICM 2018 satellite school and workshop K-theory conference in Argentina. The school was held from July 16–20, 2018, in La Plata, Argentina, and the workshop was held from July 23–27, 2018, in Buenos Aires, Argentina. The volume showcases current developments in K-theory and related areas, including motives, homological algebra, index theory, operator algebras, and their applications and connections. Papers cover topics such as K-theory of group rings, Witt groups of real algebraic varieties, coarse homology theories, topological cyclic homology, negative K-groups of monoid algebras, Milnor K-theory and regulators, noncommutative motives, the classification of C∗-algebras via Kasparov's K-theory, the comparison between full and reduced C∗-crossed products, and a proof of Bott periodicity using almost commuting matrices.

Although centered on the K-theory of operator algebras, a broad range of topics
is covered including geometric, L2- and spectral invariants, such as the analytic
torsion, signature and index, of differential and pseudo-differential operators on ...

Author: Dan Burghelea

Publisher: Springer Science & Business Media

ISBN: 3764386045

Category: Mathematics

Page: 309

View: 151

This book consists of a collection of original, refereed research and expository articles on elliptic aspects of geometric analysis on manifolds, including singular, foliated and non-commutative spaces. The topics covered include the index of operators, torsion invariants, K-theory of operator algebras and L2-invariants. There are contributions from leading specialists, and the book maintains a reasonable balance between research, expository and mixed papers.

Symptomatic of the incompleteness of this theory is that, even for AF-algebras, an
inverse functor has not yet been found ... It — what might now be called C*- algebra K -theory — became a most important tool — not only for structure and ...

Author: M. Rordam

Publisher: Springer Science & Business Media

ISBN: 3540423052

Category: Computers

Page: 198

View: 308

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.

Thus the index class is contained in the K-theory of the foliation algebra and after
pairing with the trace, an index formula is ... The best results to date on the
Novikov conjecture have been obtained using operator algebras and cyclic ...

Author: Robert S. Doran

Publisher: American Mathematical Soc.

ISBN: 9780821851753

Category: Mathematics

Page: 399

View: 121

This volume contains the proceedings of an AMS Special Session held at the Joint Mathematics Meetings in San Antonio in January 1993 to celebrate the first fifty years of $C^*$-algebra theory. The book contains carefully written expository and research articles by leaders in the field. Also included is a reprinting of the original 1943 paper on $C^*$-algebras by Gelfand and Neumark, which has had such a profound influence on the field. The volume covers a broad spectrum of topics, including the Gelfand-Neumark theorems, $C^*$-algebras and quantization, projections in $C^*$-algebras, Mackey's theory of group representations and their relation to $C^*$-algebras, transformation group $C^*$-algebras, the influence of algebraic topology on $C^*$-algebras, K-theory and index theory in operator algebras, exponential rank in $C^*$-algebras, and a survey of the development of type III von Neumann algebras. With historical perspectives and up-to-date overviews to orient readers new to the field, this book will interest mathematicians, physicists, and mathematical historians.

1 - 14 , NorthHolland , Amsterdam , 1985 ( Bld ] Blackadar , B . E . : K - Theory for Operator Algebras , Springer - Verlag , NewYork - Berlin , 1986 ( Brb ] Bourbaki ,
N . : Variétées différentielles et analytiques , Fasc . de rés . , Hermann , Paris ...

Author: Herbert Schroder

Publisher: Chapman and Hall/CRC

ISBN: UOM:39015029969386

Category: Mathematics

Page: 162

View: 353

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.