Tomita's Lectures on Observable Algebras in Hilbert Space

Tomita's Lectures on Observable Algebras in Hilbert Space

Author: Atsushi Inoue

Publisher: Springer Nature

Published: 2021-03-01

Total Pages: 197

ISBN-13: 3030688933

DOWNLOAD EBOOK

​This book is devoted to the study of Tomita's observable algebras, their structure and applications. It begins by building the foundations of the theory of T*-algebras and CT*-algebras, presenting the major results and investigating the relationship between the operator and vector representations of a CT*-algebra. It is then shown via the representation theory of locally convex*-algebras that this theory includes Tomita–Takesaki theory as a special case; every observable algebra can be regarded as an operator algebra on a Pontryagin space with codimension 1. All of the results are proved in detail and the basic theory of operator algebras on Hilbert space is summarized in an appendix. The theory of CT*-algebras has connections with many other branches of functional analysis and with quantum mechanics. The aim of this book is to make Tomita’s theory available to a wider audience, with the hope that it will be used by operator algebraists and researchers in these related fields.


Book Synopsis Tomita's Lectures on Observable Algebras in Hilbert Space by : Atsushi Inoue

Download or read book Tomita's Lectures on Observable Algebras in Hilbert Space written by Atsushi Inoue and published by Springer Nature. This book was released on 2021-03-01 with total page 197 pages. Available in PDF, EPUB and Kindle. Book excerpt: ​This book is devoted to the study of Tomita's observable algebras, their structure and applications. It begins by building the foundations of the theory of T*-algebras and CT*-algebras, presenting the major results and investigating the relationship between the operator and vector representations of a CT*-algebra. It is then shown via the representation theory of locally convex*-algebras that this theory includes Tomita–Takesaki theory as a special case; every observable algebra can be regarded as an operator algebra on a Pontryagin space with codimension 1. All of the results are proved in detail and the basic theory of operator algebras on Hilbert space is summarized in an appendix. The theory of CT*-algebras has connections with many other branches of functional analysis and with quantum mechanics. The aim of this book is to make Tomita’s theory available to a wider audience, with the hope that it will be used by operator algebraists and researchers in these related fields.

Generalized B*-Algebras and Applications

Generalized B*-Algebras and Applications

Author: Maria Fragoulopoulou

Publisher: Springer Nature

Published: 2022-06-09

Total Pages: 251

ISBN-13: 3030964337

DOWNLOAD EBOOK

This book reviews the theory of 'generalized B*-algebras' (GB*-algebras), a class of complete locally convex *-algebras which includes all C*-algebras and some of their extensions. A functional calculus and a spectral theory for GB*-algebras is presented, together with results such as Ogasawara's commutativity condition, Gelfand–Naimark type theorems, a Vidav–Palmer type theorem, an unbounded representation theory, and miscellaneous applications. Numerous contributions to the subject have been made since its initiation by G.R. Allan in 1967, including the notable early work of his student P.G. Dixon. Providing an exposition of existing research in the field, the book aims to make this growing theory as familiar as possible to postgraduate students interested in functional analysis, (unbounded) operator theory and its relationship to mathematical physics. It also addresses researchers interested in extensions of the celebrated theory of C*-algebras.


Book Synopsis Generalized B*-Algebras and Applications by : Maria Fragoulopoulou

Download or read book Generalized B*-Algebras and Applications written by Maria Fragoulopoulou and published by Springer Nature. This book was released on 2022-06-09 with total page 251 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book reviews the theory of 'generalized B*-algebras' (GB*-algebras), a class of complete locally convex *-algebras which includes all C*-algebras and some of their extensions. A functional calculus and a spectral theory for GB*-algebras is presented, together with results such as Ogasawara's commutativity condition, Gelfand–Naimark type theorems, a Vidav–Palmer type theorem, an unbounded representation theory, and miscellaneous applications. Numerous contributions to the subject have been made since its initiation by G.R. Allan in 1967, including the notable early work of his student P.G. Dixon. Providing an exposition of existing research in the field, the book aims to make this growing theory as familiar as possible to postgraduate students interested in functional analysis, (unbounded) operator theory and its relationship to mathematical physics. It also addresses researchers interested in extensions of the celebrated theory of C*-algebras.

Ten Lectures on Operator Algebras

Ten Lectures on Operator Algebras

Author: William Arveson

Publisher: American Mathematical Soc.

Published: 1984

Total Pages: 108

ISBN-13: 9780821889008

DOWNLOAD EBOOK

This book contains expanded versions of ten lectures delivered at Texas Tech University in the summer of 1983. The operator algebras of the title are nonselfadjoint algebras of operators on Hilbert space.


Book Synopsis Ten Lectures on Operator Algebras by : William Arveson

Download or read book Ten Lectures on Operator Algebras written by William Arveson and published by American Mathematical Soc.. This book was released on 1984 with total page 108 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains expanded versions of ten lectures delivered at Texas Tech University in the summer of 1983. The operator algebras of the title are nonselfadjoint algebras of operators on Hilbert space.

Tomita-Takesaki Theory in Algebras of Unbounded Operators

Tomita-Takesaki Theory in Algebras of Unbounded Operators

Author: Atsushi Inoue

Publisher: Springer

Published: 2006-11-14

Total Pages: 249

ISBN-13: 3540494952

DOWNLOAD EBOOK

These notes are devoted to a systematic study of developing the Tomita-Takesaki theory for von Neumann algebras in unbounded operator algebras called O*-algebras and to its applications to quantum physics. The notions of standard generalized vectors and standard weights for an O*-algebra are introduced and they lead to a Tomita-Takesaki theory of modular automorphisms. The Tomita-Takesaki theory in O*-algebras is applied to quantum moment problem, quantum statistical mechanics and the Wightman quantum field theory. This will be of interest to graduate students and researchers in the field of (unbounded) operator algebras and mathematical physics.


Book Synopsis Tomita-Takesaki Theory in Algebras of Unbounded Operators by : Atsushi Inoue

Download or read book Tomita-Takesaki Theory in Algebras of Unbounded Operators written by Atsushi Inoue and published by Springer. This book was released on 2006-11-14 with total page 249 pages. Available in PDF, EPUB and Kindle. Book excerpt: These notes are devoted to a systematic study of developing the Tomita-Takesaki theory for von Neumann algebras in unbounded operator algebras called O*-algebras and to its applications to quantum physics. The notions of standard generalized vectors and standard weights for an O*-algebra are introduced and they lead to a Tomita-Takesaki theory of modular automorphisms. The Tomita-Takesaki theory in O*-algebras is applied to quantum moment problem, quantum statistical mechanics and the Wightman quantum field theory. This will be of interest to graduate students and researchers in the field of (unbounded) operator algebras and mathematical physics.

Tomita's Theory of Modular Hilbert Algebras and Its Applications

Tomita's Theory of Modular Hilbert Algebras and Its Applications

Author:

Publisher:

Published: 1968

Total Pages:

ISBN-13:

DOWNLOAD EBOOK


Book Synopsis Tomita's Theory of Modular Hilbert Algebras and Its Applications by :

Download or read book Tomita's Theory of Modular Hilbert Algebras and Its Applications written by and published by . This book was released on 1968 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Tulane University Ring and Operator Theory Year, 1970-1971

Tulane University Ring and Operator Theory Year, 1970-1971

Author: Karl H. Hofmann

Publisher: Lecture Notes in Mathematics

Published: 1972-02-28

Total Pages: 804

ISBN-13:

DOWNLOAD EBOOK

From September 1970 through May 1971 Tulane University organized a special year long program in the theory of noncommutative rings and operator algebras. Visitors from various institutions of the U.S.A. and abroad contributed to a series of lectures in which they covered recent advances in their own field of specialty. These notes contain these lectures to the extent that they have not appeared elsewhere. This volume presents the lectures on operator algebras, applications to the theory of operators on Hilbert space, and harmonic analysis.


Book Synopsis Tulane University Ring and Operator Theory Year, 1970-1971 by : Karl H. Hofmann

Download or read book Tulane University Ring and Operator Theory Year, 1970-1971 written by Karl H. Hofmann and published by Lecture Notes in Mathematics. This book was released on 1972-02-28 with total page 804 pages. Available in PDF, EPUB and Kindle. Book excerpt: From September 1970 through May 1971 Tulane University organized a special year long program in the theory of noncommutative rings and operator algebras. Visitors from various institutions of the U.S.A. and abroad contributed to a series of lectures in which they covered recent advances in their own field of specialty. These notes contain these lectures to the extent that they have not appeared elsewhere. This volume presents the lectures on operator algebras, applications to the theory of operators on Hilbert space, and harmonic analysis.

Probabilistic and Statistical Aspects of Quantum Theory

Probabilistic and Statistical Aspects of Quantum Theory

Author: Alexander S. Holevo

Publisher: Springer Science & Business Media

Published: 2011-05-05

Total Pages: 324

ISBN-13: 8876423788

DOWNLOAD EBOOK

This book is devoted to aspects of the foundations of quantum mechanics in which probabilistic and statistical concepts play an essential role. The main part of the book concerns the quantitative statistical theory of quantum measurement, based on the notion of positive operator-valued measures. During the past years there has been substantial progress in this direction, stimulated to a great extent by new applications such as Quantum Optics, Quantum Communication and high-precision experiments. The questions of statistical interpretation, quantum symmetries, theory of canonical commutation relations and Gaussian states, uncertainty relations as well as new fundamental bounds concerning the accuracy of quantum measurements, are discussed in this book in an accessible yet rigorous way. Compared to the first edition, there is a new Supplement devoted to the hidden variable issue. Comments and the bibliography have also been extended and updated.


Book Synopsis Probabilistic and Statistical Aspects of Quantum Theory by : Alexander S. Holevo

Download or read book Probabilistic and Statistical Aspects of Quantum Theory written by Alexander S. Holevo and published by Springer Science & Business Media. This book was released on 2011-05-05 with total page 324 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to aspects of the foundations of quantum mechanics in which probabilistic and statistical concepts play an essential role. The main part of the book concerns the quantitative statistical theory of quantum measurement, based on the notion of positive operator-valued measures. During the past years there has been substantial progress in this direction, stimulated to a great extent by new applications such as Quantum Optics, Quantum Communication and high-precision experiments. The questions of statistical interpretation, quantum symmetries, theory of canonical commutation relations and Gaussian states, uncertainty relations as well as new fundamental bounds concerning the accuracy of quantum measurements, are discussed in this book in an accessible yet rigorous way. Compared to the first edition, there is a new Supplement devoted to the hidden variable issue. Comments and the bibliography have also been extended and updated.

Geometry of State Spaces of Operator Algebras

Geometry of State Spaces of Operator Algebras

Author: Erik M. Alfsen

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 467

ISBN-13: 1461200199

DOWNLOAD EBOOK

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.


Book Synopsis Geometry of State Spaces of Operator Algebras by : Erik M. Alfsen

Download or read book Geometry of State Spaces of Operator Algebras written by Erik M. Alfsen and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 467 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

The $mathscr {P}(varphi )_2$ Model on de Sitter Space

The $mathscr {P}(varphi )_2$ Model on de Sitter Space

Author: João C. A. Barata

Publisher: American Mathematical Society

Published: 2023-01-18

Total Pages: 282

ISBN-13: 147045548X

DOWNLOAD EBOOK

View the abstract.


Book Synopsis The $mathscr {P}(varphi )_2$ Model on de Sitter Space by : João C. A. Barata

Download or read book The $mathscr {P}(varphi )_2$ Model on de Sitter Space written by João C. A. Barata and published by American Mathematical Society. This book was released on 2023-01-18 with total page 282 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.

Deep Beauty

Deep Beauty

Author: Hans Halvorson

Publisher: Cambridge University Press

Published: 2011-04-18

Total Pages: 487

ISBN-13: 113949922X

DOWNLOAD EBOOK

No scientific theory has caused more puzzlement and confusion than quantum theory. Physics is supposed to help us to understand the world, but quantum theory makes it seem a very strange place. This book is about how mathematical innovation can help us gain deeper insight into the structure of the physical world. Chapters by top researchers in the mathematical foundations of physics explore new ideas, especially novel mathematical concepts at the cutting edge of future physics. These creative developments in mathematics may catalyze the advances that enable us to understand our current physical theories, especially quantum theory. The authors bring diverse perspectives, unified only by the attempt to introduce fresh concepts that will open up new vistas in our understanding of future physics.


Book Synopsis Deep Beauty by : Hans Halvorson

Download or read book Deep Beauty written by Hans Halvorson and published by Cambridge University Press. This book was released on 2011-04-18 with total page 487 pages. Available in PDF, EPUB and Kindle. Book excerpt: No scientific theory has caused more puzzlement and confusion than quantum theory. Physics is supposed to help us to understand the world, but quantum theory makes it seem a very strange place. This book is about how mathematical innovation can help us gain deeper insight into the structure of the physical world. Chapters by top researchers in the mathematical foundations of physics explore new ideas, especially novel mathematical concepts at the cutting edge of future physics. These creative developments in mathematics may catalyze the advances that enable us to understand our current physical theories, especially quantum theory. The authors bring diverse perspectives, unified only by the attempt to introduce fresh concepts that will open up new vistas in our understanding of future physics.