Tensor Categories And Endomorphisms Of Von Neumann Algebras
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Tensor Categories And Endomorphisms Of Von Neumann Algebras
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Author : Marcel Bischoff
language : en
Publisher: Springer
Release Date : 2015-01-13
Tensor Categories And Endomorphisms Of Von Neumann Algebras written by Marcel Bischoff and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-01-13 with Science categories.
C* tensor categories are a point of contact where Operator Algebras and Quantum Field Theory meet. They are the underlying unifying concept for homomorphisms of (properly infinite) von Neumann algebras and representations of quantum observables. The present introductory text reviews the basic notions and their cross-relations in different contexts. The focus is on Q-systems that serve as complete invariants, both for subfactors and for extensions of quantum field theory models. It proceeds with various operations on Q-systems (several decompositions, the mirror Q-system, braided product, centre and full centre of Q-systems) some of which are defined only in the presence of a braiding. The last chapter gives a brief exposition of the relevance of the mathematical structures presented in the main body for applications in Quantum Field Theory (in particular two-dimensional Conformal Field Theory, also with boundaries or defects).
Classification Of Subfactors And Their Endomorphisms
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Author : Sorin Popa
language : en
Publisher: American Mathematical Soc.
Release Date : 1995
Classification Of Subfactors And Their Endomorphisms written by Sorin Popa and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995 with Mathematics categories.
This monograph provides a more unified and self-contained presentation of the results presented in Popa's earlier papers on this topic. The classification is in terms of the standard invariant $\mathcal G_{\mathcal N,\mathcal M}$ of the subfactor $\mathcal N\subset \mathcal M$. This invariant is a lattice of inclusions of finite dimensional algebras associated with the Jones iterated basic construction for $\mathcal N\subset \mathcal M$. ``Classification of Subfactors and Their Endomorphisms'' is based on lectures presented by Popa at the NSF-CBMS Regional Conference held in Eugene, Oregon, in August 1993.
Classification Of Nuclear C Algebras Entropy In Operator Algebras
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Author : M. Rordam
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-04-18
Classification Of Nuclear C Algebras Entropy In Operator Algebras written by M. Rordam and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-04-18 with Mathematics categories.
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.
Tensor Categories
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Author : Pavel Etingof
language : en
Publisher: American Mathematical Soc.
Release Date : 2016-08-05
Tensor Categories written by Pavel Etingof and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-08-05 with Algebraic topology categories.
Is there a vector space whose dimension is the golden ratio? Of course not—the golden ratio is not an integer! But this can happen for generalizations of vector spaces—objects of a tensor category. The theory of tensor categories is a relatively new field of mathematics that generalizes the theory of group representations. It has deep connections with many other fields, including representation theory, Hopf algebras, operator algebras, low-dimensional topology (in particular, knot theory), homotopy theory, quantum mechanics and field theory, quantum computation, theory of motives, etc. This book gives a systematic introduction to this theory and a review of its applications. While giving a detailed overview of general tensor categories, it focuses especially on the theory of finite tensor categories and fusion categories (in particular, braided and modular ones), and discusses the main results about them with proofs. In particular, it shows how the main properties of finite-dimensional Hopf algebras may be derived from the theory of tensor categories. Many important results are presented as a sequence of exercises, which makes the book valuable for students and suitable for graduate courses. Many applications, connections to other areas, additional results, and references are discussed at the end of each chapter.
Actions Of Discrete Amenable Groups On Von Neumann Algebras
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Author : Adrian Ocneanu
language : en
Publisher: Springer
Release Date : 2006-12-08
Actions Of Discrete Amenable Groups On Von Neumann Algebras written by Adrian Ocneanu and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-12-08 with Mathematics categories.
Operator Algebras And Applications
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Author : A. Katavolos
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Operator Algebras And Applications written by A. Katavolos and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
During the last few years, the theory of operator algebras, particularly non-self-adjoint operator algebras, has evolved dramatically, experiencing both international growth and interfacing with other important areas. The present volume presents a survey of some of the latest developments in the field in a form that is detailed enough to be accessible to advanced graduate students as well as researchers in the field. Among the topics treated are: operator spaces, Hilbert modules, limit algebras, reflexive algebras and subspaces, relations to basis theory, C* algebraic quantum groups, endomorphisms of operator algebras, conditional expectations and projection maps, and applications, particularly to wavelet theory. The volume also features an historical paper offering a new approach to the Pythagoreans' discovery of irrational numbers.
Operator Algebras
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Author : Bruce Blackadar
language : en
Publisher: Taylor & Francis
Release Date : 2006
Operator Algebras written by Bruce Blackadar and has been published by Taylor & Francis this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006 with Mathematics categories.
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.
Conformal Field Theories And Tensor Categories
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Author : Chengming Bai
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-10-30
Conformal Field Theories And Tensor Categories written by Chengming Bai and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-10-30 with Mathematics categories.
The present volume is a collection of seven papers that are either based on the talks presented at the workshop "Conformal field theories and tensor categories" held June 13 to June 17, 2011 at the Beijing International Center for Mathematical Research, Peking University, or are extensions of the material presented in the talks at the workshop. These papers present new developments beyond rational conformal field theories and modular tensor categories and new applications in mathematics and physics. The topics covered include tensor categories from representation categories of Hopf algebras, applications of conformal field theories and tensor categories to topological phases and gapped systems, logarithmic conformal field theories and the corresponding non-semisimple tensor categories, and new developments in the representation theory of vertex operator algebras. Some of the papers contain detailed introductory material that is helpful for graduate students and researchers looking for an introduction to these research directions. The papers also discuss exciting recent developments in the area of conformal field theories, tensor categories and their applications and will be extremely useful for researchers working in these areas.
Topological Phases Of Matter And Quantum Computation
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Author : Paul Bruillard
language : en
Publisher: American Mathematical Soc.
Release Date : 2020-03-31
Topological Phases Of Matter And Quantum Computation written by Paul Bruillard and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-03-31 with Education categories.
This volume contains the proceedings of the AMS Special Session on Topological Phases of Matter and Quantum Computation, held from September 24–25, 2016, at Bowdoin College, Brunswick, Maine. Topological quantum computing has exploded in popularity in recent years. Sitting at the triple point between mathematics, physics, and computer science, it has the potential to revolutionize sub-disciplines in these fields. The academic importance of this field has been recognized in physics through the 2016 Nobel Prize. In mathematics, some of the 1990 Fields Medals were awarded for developments in topics that nowadays are fundamental tools for the study of topological quantum computation. Moreover, the practical importance of this discipline has been underscored by recent industry investments. The relative youth of this field combined with a high degree of interest in it makes now an excellent time to get involved. Furthermore, the cross-disciplinary nature of topological quantum computing provides an unprecedented number of opportunities for cross-pollination of mathematics, physics, and computer science. This can be seen in the variety of works contained in this volume. With articles coming from mathematics, physics, and computer science, this volume aims to provide a taste of different sub-disciplines for novices and a wealth of new perspectives for veteran researchers. Regardless of your point of entry into topological quantum computing or your experience level, this volume has something for you.
Lectures On Von Neumann Algebras
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Author : Șerban Strătilă
language : en
Publisher: Cambridge University Press
Release Date : 2019-05-09
Lectures On Von Neumann Algebras written by Șerban Strătilă and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-05-09 with Mathematics categories.
The text covers fundamentals of von Neumann algebras, including the Tomita's theory of von Neumann algebras and the latest developments.
