Author: Sh. Ayupov,Abdugafur Rakhimov,Shukhrat Usmanov

Publisher: Springer Science & Business Media

ISBN: 9401586055

Category: Mathematics

Page: 230

View: 8012

Author: Sh. Ayupov,Abdugafur Rakhimov,Shukhrat Usmanov

Publisher: Springer Science & Business Media

ISBN: 9401586055

Category: Mathematics

Page: 230

View: 8012

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: Erik M. Alfsen,Frederic W. Shultz

Publisher: Springer Science & Business Media

ISBN: 1461200199

Category: Mathematics

Page: 467

View: 7852

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: Constantin Vârsan

Publisher: Springer Science & Business Media

ISBN: 9401146799

Category: Mathematics

Page: 243

View: 1764

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.

*International Conference, Algebra and Its Applications, March 22-26, 2005, Ohio University, Athens, Ohio*

Author: D. V. Huynh,Surender Kumar Jain,Sergio R. López-Permouth

Publisher: American Mathematical Soc.

ISBN: 082181950X

Category: Mathematics

Page: 569

View: 4152

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: Miguel Cabrera García,Ángel Rodríguez Palacios

Publisher: Cambridge University Press

ISBN: 1139992775

Category: Mathematics

Page: N.A

View: 3949

This first systematic account of the basic theory of normed algebras, without assuming associativity, includes many new and unpublished results and is sure to become a central resource for researchers and graduate students in the field. This first volume focuses on the non-associative generalizations of (associative) C*-algebras provided by the so-called non-associative Gelfand–Naimark and Vidav–Palmer theorems, which give rise to alternative C*-algebras and non-commutative JB*-algebras, respectively. The relationship between non-commutative JB*-algebras and JB*-triples is also fully discussed. The second volume covers Zel'manov's celebrated work in Jordan theory to derive classification theorems for non-commutative JB*-algebras and JB*-triples, as well as other topics. The book interweaves pure algebra, geometry of normed spaces, and complex analysis, and includes a wealth of historical comments, background material, examples and exercises. The authors also provide an extensive bibliography.

Author: Miguel Cabrera García,Ángel Rodríguez Palacios

Publisher: Cambridge University Press

ISBN: 1108631436

Category: Mathematics

Page: 760

View: 2121

This first systematic account of the basic theory of normed algebras, without assuming associativity, includes many new and unpublished results and is sure to become a central resource for researchers and graduate students in the field. This second volume revisits JB*-triples, covers Zel'manov's celebrated work in Jordan theory, proves the unit-free variant of the Vidav–Palmer theorem, and develops the representation theory of alternative C*-algebras and non-commutative JB*-algebras. This completes the work begun in the first volume, which introduced these algebras and discussed the so-called non-associative Gelfand–Naimark and Vidav–Palmer theorems. This book interweaves pure algebra, geometry of normed spaces, and infinite-dimensional complex analysis. Novel proofs are presented in complete detail at a level accessible to graduate students. The book contains a wealth of historical comments, background material, examples, and an extensive bibliography.

Author: Miguel Cabrera García,Ángel Rodríguez Palacios

Publisher: Cambridge University Press

ISBN: 1107043115

Category: Mathematics

Page: 760

View: 4986

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

Author: Kunio Murasugi,B. Kurpita

Publisher: Springer Science & Business Media

ISBN: 9780792357674

Category: Computers

Page: 272

View: 1646

This book provides a comprehensive exposition of the theory of braids, beginning with the basic mathematical definitions and structures. Among the many topics explained in detail are: the braid group for various surfaces; the solution of the word problem for the braid group; braids in the context of knots and links (Alexander's theorem); Markov's theorem and its use in obtaining braid invariants; the connection between the Platonic solids (regular polyhedra) and braids; the use of braids in the solution of algebraic equations. Dirac's problem and special types of braids termed Mexican plaits are also discussed. Audience: Since the book relies on concepts and techniques from algebra and topology, the authors also provide a couple of appendices that cover the necessary material from these two branches of mathematics. Hence, the book is accessible not only to mathematicians but also to anybody who might have an interest in the theory of braids. In particular, as more and more applications of braid theory are found outside the realm of mathematics, this book is ideal for any physicist, chemist or biologist who would like to understand the mathematics of braids. With its use of numerous figures to explain clearly the mathematics, and exercises to solidify the understanding, this book may also be used as a textbook for a course on knots and braids, or as a supplementary textbook for a course on topology or algebra.

*Theory of C*-Algebras and von Neumann Algebras*

Author: Bruce Blackadar

Publisher: Springer Science & Business Media

ISBN: 3540285172

Category: Mathematics

Page: 528

View: 5362

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: Kevin McCrimmon

Publisher: Springer Science & Business Media

ISBN: 0387217967

Category: Mathematics

Page: 563

View: 4541

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: Wolfgang Bertram

Publisher: Springer Science & Business Media

ISBN: 3540414266

Category: Mathematics

Page: 269

View: 6518

0. In this work of we the Lie- and Jordan on an study interplay theory and ona level.Weintendtocontinue ittoa algebraic geometric systematicstudy ofthe role Jordan inharmonic In the of theoryplays analysis. fact, applications the of Jordan to theharmonic on cones theory algebras analysis symmetric (cf. of the wereatthe theauthor'sworkinthisarea. Then monograph[FK94]) origin Jordan in of turned the causal algebras up study many symmetric (see spaces Section and clearthat all soon itbecame XI.3), "generically" symmetric spaces have Since a relation toJordan Jordan does not significant theory. theory (yet) to the standard tools inharmonic the is text belong analysis, present designed to self-contained introduction to Jordan for readers a provide theory having basic Lie and Our ofview some on knowledge groups symmetric spaces. point is introduce first the relevant structures geometric: throughout we geometric anddeducefromtheir identities fortheassociated propertiesalgebraic algebraic structures. Thus our differs from related ones presentation (cf. e.g. [FK94], the fact thatwe do not take an axiomatic definition ofsome [Lo77], [Sa80]) by Jordan structureasour Let us nowanoverviewof algebraic startingpoint. give the See alsothe introductions the contents. at ofeach given beginning chapter. 0.1. Lie and Jordan Ifwe the associative algebras algebras. decompose of the matrix in its and product algebra M(n, R) symmetric skew-symmetric parts, - XY YX XY YX + XY= + (0.1) 2 2 then second the term leads to the Lie with algebra gf(n, R) product [X, Y] XY- and first the termleadstotheJordan M with YX, algebra (n, R) product - X Y= + (XY YX).

*their representations, generalisations, and applications*

Author: Boris Petrovich Komrakov

Publisher: Kluwer Academic Pub

ISBN: 9780792349167

Category: Mathematics

Page: 442

View: 5655

This collection brings together papers related to the classical ideas of Sophus Lie. The present work reflects the interests of scientists associated with the International Sophus Lie Center, and provides up-to-date results in Lie groups and Lie algebras, quantum mathematics, hypergroups, homogeneous spaces, Lie superalgebras, the theory of representations and applications to differential equations and integrable systems. Among the topics that are treated are quantization of Poisson structures, applications of multivalued groups, noncommutative aspects of hypergroups, homology invariants of homogeneous spaces, generalisations of the Godbillon-Vey invariant, relations between classical problems of linear analysis and representation theory and the geometry of current groups. Audience: This volume will be of interest to mathematicians and physicists specialising in the theory and applications of Lie groups and Lie algebras, quantum groups, hypergroups and homogeneous spaces.

Author: Lev Sabinin,Larissa Sbitneva,Ivan Shestakov

Publisher: CRC Press

ISBN: 9780824726690

Category: Mathematics

Page: 552

View: 1852

With contributions derived from presentations at an international conference, Non-Associative Algebra and Its Applications explores a wide range of topics focusing on Lie algebras, nonassociative rings and algebras, quasigroups, loops, and related systems as well as applications of nonassociative algebra to geometry, physics, and natural sciences. This book covers material such as Jordan superalgebras, nonassociative deformations, nonassociative generalization of Hopf algebras, the structure of free algebras, derivations of Lie algebras, and the identities of Albert algebra. It also includes applications of smooth quasigroups and loops to differential geometry and relativity.

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Mathematics

Page: N.A

View: 5731

Author: Harald Hanche-Olsen,Erling Størmer

Publisher: Pitman Advanced Publishing Program

ISBN: N.A

Category: Mathematics

Page: 183

View: 3274

*Classical and Quantum Aspects*

Author: A.K. Prykarpatsky,I.V. Mykytiuk

Publisher: Springer

ISBN: 9780792350903

Category: Science

Page: 559

View: 2035

In recent times it has been stated that many dynamical systems of classical mathematical physics and mechanics are endowed with symplectic structures, given in the majority of cases by Poisson brackets. Very often such Poisson structures on corresponding manifolds are canonical, which gives rise to the possibility of producing their hidden group theoretical essence for many completely integrable dynamical systems. It is a well understood fact that great part of comprehensive integrability theories of nonlinear dynamical systems on manifolds is based on Lie-algebraic ideas, by means of which, in particular, the classification of such compatibly bi Hamiltonian and isospectrally Lax type integrable systems has been carried out. Many chapters of this book are devoted to their description, but to our regret so far the work has not been completed. Hereby our main goal in each analysed case consists in separating the basic algebraic essence responsible for the complete integrability, and which is, at the same time, in some sense universal, i. e. , characteristic for all of them. Integrability analysis in the framework of a gradient-holonomic algorithm, devised in this book, is fulfilled through three stages: 1) finding a symplectic structure (Poisson bracket) transforming an original dynamical system into a Hamiltonian form; 2) finding first integrals (action variables or conservation laws); 3) defining an additional set of variables and some functional operator quantities with completely controlled evolutions (for instance, as Lax type representation).

*Proceedings of the International Conference on Mathematical Analysis and its Applications, Kuwait, 1985*

Author: S. M. Mazhar,A. Hamoui,N. S. Faour

Publisher: Elsevier

ISBN: 1483148114

Category: Mathematics

Page: 442

View: 9152

Mathematical Analysis and its Applications covers the proceedings of the International Conference on Mathematical Analysis and its Applications. The book presents studies that discuss several mathematical analysis methods and their respective applications. The text presents 38 papers that discuss topics, such as approximation of continuous functions by ultraspherical series and classes of bi-univalent functions. The representation of multipliers of eigen and joint function expansions of nonlocal spectral problems for first- and second-order differential operators is also discussed. The book will be of great interest to researchers and professionals whose work involves the use of mathematical analysis.

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Mathematicians

Page: N.A

View: 4486

*A Unified Approach*

Author: Gershon Kresin,Vladimir Maz'ya

Publisher: Springer

ISBN: 3540695745

Category: Mathematics

Page: 145

View: 9751

This volume contains a coherent point of view on various sharp pointwise inequalities for analytic functions in a disk in terms of the real part of the function on the boundary circle or in the disk itself. Inequalities of this type are frequently used in the theory of entire functions and in the analytic number theory.

*Theory and Applications*

Author: V.I. Istratescu

Publisher: Springer Science & Business Media

ISBN: 9789027721822

Category: Mathematics

Page: 896

View: 6207

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Oad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.