Jordan Real And Lie Structures In Operator Algebras

Author : Sh. Ayupov,Abdugafur Rakhimov,Shukhrat Usmanov
Publisher : Springer Science & Business Media
Page : 239 pages
File Size : 40,7 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9789401586054

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Jordan, Real and Lie Structures in Operator Algebras by Sh. Ayupov,Abdugafur Rakhimov,Shukhrat Usmanov Pdf

The theory of operator algebras acting on a Hilbert space was initiated in thirties by papers of Murray and von Neumann. In these papers they have studied the structure of algebras which later were called von Neu mann algebras or W* -algebras. They are weakly closed complex *-algebras of operators on a Hilbert space. At present the theory of von Neumann algebras is a deeply developed theory with various applications. In the framework of von Neumann algebras theory the study of fac tors (i.e. W* -algebras with trivial centres) is very important, since they are comparatively simple and investigation of general W* -algebras can be reduced to the case of factors. Therefore the theory of factors is one of the main tools in the structure theory of von Neumann algebras. In the middle of sixtieth Topping [To 1] and Stormer [S 2] have ini tiated the study of Jordan (non associative and real) analogues of von Neumann algebras - so called JW-algebras, i.e. real linear spaces of self adjoint opera.tors on a complex Hilbert space, which contain the identity operator 1. closed with respect to the Jordan (i.e. symmetrised) product INTRODUCTION 2 x 0 y = ~(Xy + yx) and closed in the weak operator topology. The structure of these algebras has happened to be close to the struc ture of von Neumann algebras and it was possible to apply ideas and meth ods similar to von Neumann algebras theory in the study of JW-algebras.

Author : Wolfgang Bertram
Publisher : Springer
Page : 274 pages
File Size : 47,8 Mb
Release : 2003-07-01
Category : Mathematics
ISBN : 9783540444589

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The Geometry of Jordan and Lie Structures by Wolfgang Bertram Pdf

The geometry of Jordan and Lie structures tries to answer the following question: what is the integrated, or geometric, version of real Jordan algebras, - triple systems and - pairs? Lie theory shows the way one has to go: Lie groups and symmetric spaces are the geometric version of Lie algebras and Lie triple systems. It turns out that both geometries are closely related via a functor between them, called the Jordan-Lie functor, which is constructed in this book. The reader is not assumed to have any knowledge of Jordan theory; the text can serve as a self-contained introduction to (real finite-dimensional) Jordan theory.

Author : Antonio Fernández López
Publisher : American Mathematical Soc.
Page : 314 pages
File Size : 47,9 Mb
Release : 2019-08-19
Category : Jordan algebras
ISBN : 9781470450861

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Jordan Structures in Lie Algebras by Antonio Fernández López Pdf

Explores applications of Jordan theory to the theory of Lie algebras. After presenting the general theory of nonassociative algebras and of Lie algebras, the book then explains how properties of the Jordan algebra attached to a Jordan element of a Lie algebra can be used to reveal properties of the Lie algebra itself.

Author : Erik M. Alfsen,Frederic W. Shultz
Publisher : Springer Science & Business Media
Page : 467 pages
File Size : 41,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461200192

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Geometry of State Spaces of Operator Algebras by Erik M. Alfsen,Frederic W. Shultz Pdf

In this book we give a complete geometric description of state spaces of operator algebras, Jordan as well as associative. That is, we give axiomatic characterizations of those convex sets that are state spaces of C*-algebras and von Neumann algebras, together with such characterizations for the normed Jordan algebras called JB-algebras and JBW-algebras. These non associative algebras generalize C*-algebras and von Neumann algebras re spectively, and the characterization of their state spaces is not only of interest in itself, but is also an important intermediate step towards the characterization of the state spaces of the associative algebras. This book gives a complete and updated presentation of the character ization theorems of [10]' [11] and [71]. Our previous book State spaces of operator algebras: basic theory, orientations and C*-products, referenced as [AS] in the sequel, gives an account of the necessary prerequisites on C*-algebras and von Neumann algebras, as well as a discussion of the key notion of orientations of state spaces. For the convenience of the reader, we have summarized these prerequisites in an appendix which contains all relevant definitions and results (listed as (AI), (A2), ... ), with reference back to [AS] for proofs, so that this book is self-contained.

Author : Harald Upmeier
Publisher : American Mathematical Soc.
Page : 100 pages
File Size : 43,6 Mb
Release : 1987-01-01
Category : Mathematics
ISBN : 0821889125

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Jordan Algebras in Analysis, Operator Theory, and Quantum Mechanics by Harald Upmeier Pdf

Author : Cho-Ho Chu
Publisher : Cambridge University Press
Page : 273 pages
File Size : 40,7 Mb
Release : 2011-11-17
Category : Mathematics
ISBN : 9781139505437

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Jordan Structures in Geometry and Analysis by Cho-Ho Chu Pdf

Jordan theory has developed rapidly in the last three decades, but very few books describe its diverse applications. Here, the author discusses some recent advances of Jordan theory in differential geometry, complex and functional analysis, with the aid of numerous examples and concise historical notes. These include: the connection between Jordan and Lie theory via the Tits–Kantor–Koecher construction of Lie algebras; a Jordan algebraic approach to infinite dimensional symmetric manifolds including Riemannian symmetric spaces; the one-to-one correspondence between bounded symmetric domains and JB*-triples; and applications of Jordan methods in complex function theory. The basic structures and some functional analytic properties of JB*-triples are also discussed. The book is a convenient reference for experts in complex geometry or functional analysis, as well as an introduction to these areas for beginning researchers. The recent applications of Jordan theory discussed in the book should also appeal to algebraists.

Author : Dinh Van Huynh,International conference on algebra and its applications,Surender Kumar Jain
Publisher : American Mathematical Soc.
Page : 586 pages
File Size : 47,9 Mb
Release : 2000
Category : Algebra
ISBN : 9780821819500

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Algebra and Its Applications by Dinh Van Huynh,International conference on algebra and its applications,Surender Kumar Jain Pdf

Among all areas of mathematics, algebra is one of the best suited to find applications within the frame of our booming technological society. The thirty-eight articles in this volume encompass the proceedings of the International Conference on Algebra and Its Applications (Athens, OH, 1999), which explored the applications and interplay among the disciplines of ring theory, linear algebra, and coding theory. The presentations collected here reflect the dialogue between mathematicians involved in theoretical aspects of algebra and mathematicians involved in solving problems where state-of-the-art research tools may be used and applied. This Contemporary Mathematics series volume communicates the potential for collaboration among those interested in exploring the wealth of applications for abstract algebra in fields such as information and coding. The expository papers would serve well as supplemental reading in graduate seminars.

Author : Constantin Vârsan
Publisher : Springer Science & Business Media
Page : 243 pages
File Size : 52,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789401146791

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Applications of Lie Algebras to Hyperbolic and Stochastic Differential Equations by Constantin Vârsan Pdf

The main part of the book is based on a one semester graduate course for students in mathematics. I have attempted to develop the theory of hyperbolic systems of differen tial equations in a systematic way, making as much use as possible ofgradient systems and their algebraic representation. However, despite the strong sim ilarities between the development of ideas here and that found in a Lie alge bras course this is not a book on Lie algebras. The order of presentation has been determined mainly by taking into account that algebraic representation and homomorphism correspondence with a full rank Lie algebra are the basic tools which require a detailed presentation. I am aware that the inclusion of the material on algebraic and homomorphism correspondence with a full rank Lie algebra is not standard in courses on the application of Lie algebras to hyperbolic equations. I think it should be. Moreover, the Lie algebraic structure plays an important role in integral representation for solutions of nonlinear control systems and stochastic differential equations yelding results that look quite different in their original setting. Finite-dimensional nonlin ear filters for stochastic differential equations and, say, decomposability of a nonlinear control system receive a common understanding in this framework.

Author : Miguel Cabrera García,Ángel Rodríguez Palacios
Publisher : Cambridge University Press
Page : 735 pages
File Size : 54,9 Mb
Release : 2014-07-31
Category : Mathematics
ISBN : 9781107043060

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Non-Associative Normed Algebras by Miguel Cabrera García,Ángel Rodríguez Palacios Pdf

The first systematic account of the basic theory of normed algebras, without assuming associativity. Sure to become a central resource.

Author : Daniel Beltita,Mihai Sabac
Publisher : Birkhäuser
Page : 226 pages
File Size : 54,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034883320

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Lie Algebras of Bounded Operators by Daniel Beltita,Mihai Sabac Pdf

In several proofs from the theory of finite-dimensional Lie algebras, an essential contribution comes from the Jordan canonical structure of linear maps acting on finite-dimensional vector spaces. On the other hand, there exist classical results concerning Lie algebras which advise us to use infinite-dimensional vector spaces as well. For example, the classical Lie Theorem asserts that all finite-dimensional irreducible representations of solvable Lie algebras are one-dimensional. Hence, from this point of view, the solvable Lie algebras cannot be distinguished from one another, that is, they cannot be classified. Even this example alone urges the infinite-dimensional vector spaces to appear on the stage. But the structure of linear maps on such a space is too little understood; for these linear maps one cannot speak about something like the Jordan canonical structure of matrices. Fortunately there exists a large class of linear maps on vector spaces of arbi trary dimension, having some common features with the matrices. We mean the bounded linear operators on a complex Banach space. Certain types of bounded operators (such as the Dunford spectral, Foia§ decomposable, scalar generalized or Colojoara spectral generalized operators) actually even enjoy a kind of Jordan decomposition theorem. One of the aims of the present book is to expound the most important results obtained until now by using bounded operators in the study of Lie algebras.

Author : Kevin McCrimmon
Publisher : Springer Science & Business Media
Page : 563 pages
File Size : 54,7 Mb
Release : 2006-05-29
Category : Mathematics
ISBN : 9780387217963

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A Taste of Jordan Algebras by Kevin McCrimmon Pdf

This book describes the history of Jordan algebras and describes in full mathematical detail the recent structure theory for Jordan algebras of arbitrary dimension due to Efim Zel'manov. Jordan algebras crop up in many surprising settings, and find application to a variety of mathematical areas. No knowledge is required beyond standard first-year graduate algebra courses.

Author : Jonathan S. Golan
Publisher : Springer Science & Business Media
Page : 207 pages
File Size : 45,7 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9789401592413

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Power Algebras over Semirings by Jonathan S. Golan Pdf

This monograph is a continuation of several themes presented in my previous books [146, 149]. In those volumes, I was concerned primarily with the properties of semirings. Here, the objects of investigation are sets of the form RA, where R is a semiring and A is a set having a certain structure. The problem is one of translating that structure to RA in some "natural" way. As such, it tries to find a unified way of dealing with diverse topics in mathematics and theoretical com puter science as formal language theory, the theory of fuzzy algebraic structures, models of optimal control, and many others. Another special case is the creation of "idempotent analysis" and similar work in optimization theory. Unlike the case of the previous work, which rested on a fairly established mathematical foundation, the approach here is much more tentative and docimastic. This is an introduction to, not a definitative presentation of, an area of mathematics still very much in the making. The basic philosphical problem lurking in the background is one stated suc cinctly by Hahle and Sostak [185]: ". . . to what extent basic fields of mathematics like algebra and topology are dependent on the underlying set theory?" The conflicting definitions proposed by various researchers in search of a resolution to this conundrum show just how difficult this problem is to see in a proper light.

Author : B.P. Komrakov,I.S. Krasil'shchik,G.L. Litvinov,A.B. Sossinsky
Publisher : Springer Science & Business Media
Page : 442 pages
File Size : 40,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789401152587

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Lie Groups and Lie Algebras by B.P. Komrakov,I.S. Krasil'shchik,G.L. Litvinov,A.B. Sossinsky Pdf

This collection contains papers conceptually related to the classical ideas of Sophus Lie (i.e., to Lie groups and Lie algebras). Obviously, it is impos sible to embrace all such topics in a book of reasonable size. The contents of this one reflect the scientific interests of those authors whose activities, to some extent at least, are associated with the International Sophus Lie Center. We have divided the book into five parts in accordance with the basic topics of the papers (although it can be easily seen that some of them may be attributed to several parts simultaneously). The first part (quantum mathematics) combines the papers related to the methods generated by the concepts of quantization and quantum group. The second part is devoted to the theory of hypergroups and Lie hypergroups, which is one of the most important generalizations of the classical concept of locally compact group and of Lie group. A natural harmonic analysis arises on hypergroups, while any abstract transformation of Fourier type is gen erated by some hypergroup (commutative or not). Part III contains papers on the geometry of homogeneous spaces, Lie algebras and Lie superalgebras. Classical problems of the representation theory for Lie groups, as well as for topological groups and semigroups, are discussed in the papers of Part IV. Finally, the last part of the collection relates to applications of the ideas of Sophus Lie to differential equations.

Author : Matej Brešar,Mikhail A. Chebotar,Wallace S. Martindale
Publisher : Springer Science & Business Media
Page : 274 pages
File Size : 41,9 Mb
Release : 2007-08-08
Category : Mathematics
ISBN : 9783764377960

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Functional Identities by Matej Brešar,Mikhail A. Chebotar,Wallace S. Martindale Pdf

A functional identity can be informally described as an identical relation involving arbitrary elements in an associative ring together with arbitrary (unknown) functions. The theory of functional identities is a relatively new one, and this is the first book on this subject. The book is accessible to a wide audience and touches on a variety of mathematical areas such as ring theory, algebra and operator theory.

Author : Pere Ara,Martin Mathieu
Publisher : Springer Science & Business Media
Page : 319 pages
File Size : 44,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781447100454

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Local Multipliers of C*-Algebras by Pere Ara,Martin Mathieu Pdf

Many problems in operator theory lead to the consideration ofoperator equa tions, either directly or via some reformulation. More often than not, how ever, the underlying space is too 'small' to contain solutions of these equa tions and thus it has to be 'enlarged' in some way. The Berberian-Quigley enlargement of a Banach space, which allows one to convert approximate into genuine eigenvectors, serves as a classical example. In the theory of operator algebras, a C*-algebra A that turns out to be small in this sense tradition ally is enlarged to its (universal) enveloping von Neumann algebra A". This works well since von Neumann algebras are in many respects richer and, from the Banach space point of view, A" is nothing other than the second dual space of A. Among the numerous fruitful applications of this principle is the well-known Kadison-Sakai theorem ensuring that every derivation 8 on a C*-algebra A becomes inner in A", though 8 may not be inner in A. The transition from A to A" however is not an algebraic one (and cannot be since it is well known that the property of being a von Neumann algebra cannot be described purely algebraically). Hence, ifthe C*-algebra A is small in an algebraic sense, say simple, it may be inappropriate to move on to A". In such a situation, A is typically enlarged by its multiplier algebra M(A).