An Introduction to Operator Algebras

An Introduction to Operator Algebras
Author: Kehe Zhu
Publisher: CRC Press
Total Pages: 172
Release: 1993-05-27
Genre: Mathematics
ISBN: 9780849378751

An Introduction to Operator Algebras is a concise text/reference that focuses on the fundamental results in operator algebras. Results discussed include Gelfand's representation of commutative C*-algebras, the GNS construction, the spectral theorem, polar decomposition, von Neumann's double commutant theorem, Kaplansky's density theorem, the (continuous, Borel, and L8) functional calculus for normal operators, and type decomposition for von Neumann algebras. Exercises are provided after each chapter.

Theory of Operator Algebras I

Theory of Operator Algebras I
Author: Masamichi Takesaki
Publisher: Springer Science & Business Media
Total Pages: 424
Release: 2012-12-06
Genre: Mathematics
ISBN: 1461261880

Mathematics for infinite dimensional objects is becoming more and more important today both in theory and application. Rings of operators, renamed von Neumann algebras by J. Dixmier, were first introduced by J. von Neumann fifty years ago, 1929, in [254] with his grand aim of giving a sound founda tion to mathematical sciences of infinite nature. J. von Neumann and his collaborator F. J. Murray laid down the foundation for this new field of mathematics, operator algebras, in a series of papers, [240], [241], [242], [257] and [259], during the period of the 1930s and early in the 1940s. In the introduction to this series of investigations, they stated Their solution 1 {to the problems of understanding rings of operators) seems to be essential for the further advance of abstract operator theory in Hilbert space under several aspects. First, the formal calculus with operator-rings leads to them. Second, our attempts to generalize the theory of unitary group-representations essentially beyond their classical frame have always been blocked by the unsolved questions connected with these problems. Third, various aspects of the quantum mechanical formalism suggest strongly the elucidation of this subject. Fourth, the knowledge obtained in these investigations gives an approach to a class of abstract algebras without a finite basis, which seems to differ essentially from all types hitherto investigated. Since then there has appeared a large volume of literature, and a great deal of progress has been achieved by many mathematicians.

Operator Algebras

Operator Algebras
Author: Bruce Blackadar
Publisher: Taylor & Francis
Total Pages: 552
Release: 2006
Genre: Mathematics
ISBN: 9783540284864

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.

Introduction to Vertex Operator Algebras and Their Representations

Introduction to Vertex Operator Algebras and Their Representations
Author: James Lepowsky
Publisher: Springer Science & Business Media
Total Pages: 330
Release: 2012-12-06
Genre: Mathematics
ISBN: 0817681868

* Introduces the fundamental theory of vertex operator algebras and its basic techniques and examples. * Begins with a detailed presentation of the theoretical foundations and proceeds to a range of applications. * Includes a number of new, original results and brings fresh perspective to important works of many other researchers in algebra, lie theory, representation theory, string theory, quantum field theory, and other areas of math and physics.

Fundamentals of the Theory of Operator Algebras. Volume III

Fundamentals of the Theory of Operator Algebras. Volume III
Author: Richard V. Kadison
Publisher: American Mathematical Soc.
Total Pages: 290
Release: 1998-01-13
Genre: Mathematics
ISBN: 0821894692

This volume is the companion volume to Fundamentals of the Theory of Operator Algebras. Volume I--Elementary Theory (Graduate Studies in Mathematics series, Volume 15). 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 I--Elementary Theory.

Introduction to Operator Space Theory

Introduction to Operator Space Theory
Author: Gilles Pisier
Publisher: Cambridge University Press
Total Pages: 492
Release: 2003-08-25
Genre: Mathematics
ISBN: 9780521811651

An introduction to the theory of operator spaces, emphasising applications to C*-algebras.

C*-Algebras and Operator Theory

C*-Algebras and Operator Theory
Author: Gerald J. Murphy
Publisher: Academic Press
Total Pages: 297
Release: 2014-06-28
Genre: Mathematics
ISBN: 0080924964

This book constitutes a first- or second-year graduate course in operator theory. It is a field that has great importance for other areas of mathematics and physics, such as algebraic topology, differential geometry, and quantum mechanics. It assumes a basic knowledge in functional analysis but no prior acquaintance with operator theory is required.

Vertex Operator Algebras and the Monster

Vertex Operator Algebras and the Monster
Author: Igor Frenkel
Publisher: Academic Press
Total Pages: 563
Release: 1989-05-01
Genre: Mathematics
ISBN: 0080874541

This work is motivated by and develops connections between several branches of mathematics and physics--the theories of Lie algebras, finite groups and modular functions in mathematics, and string theory in physics. The first part of the book presents a new mathematical theory of vertex operator algebras, the algebraic counterpart of two-dimensional holomorphic conformal quantum field theory. The remaining part constructs the Monster finite simple group as the automorphism group of a very special vertex operator algebra, called the "moonshine module" because of its relevance to "monstrous moonshine."

State Spaces of Operator Algebras

State Spaces of Operator Algebras
Author: Erik M. Alfsen
Publisher: Springer Science & Business Media
Total Pages: 372
Release: 2001-04-27
Genre: Mathematics
ISBN: 9780817638900

The topic of this book is the theory of state spaces of operator algebras and their geometry. The states are of interest because they determine representations of the algebra, and its algebraic structure is in an intriguing and fascinating fashion encoded in the geometry of the state space. From the beginning the theory of operator algebras was motivated by applications to physics, but recently it has found unexpected new applica tions to various fields of pure mathematics, like foliations and knot theory, and (in the Jordan algebra case) also to Banach manifolds and infinite di mensional holomorphy. This makes it a relevant field of study for readers with diverse backgrounds and interests. Therefore this book is not intended solely for specialists in operator algebras, but also for graduate students and mathematicians in other fields who want to learn the subject. We assume that the reader starts out with only the basic knowledge taught in standard graduate courses in real and complex variables, measure theory and functional analysis. We have given complete proofs of basic results on operator algebras, so that no previous knowledge in this field is needed. For discussion of some topics, more advanced prerequisites are needed. Here we have included all necessary definitions and statements of results, but in some cases proofs are referred to standard texts. In those cases we have tried to give references to material that can be read and understood easily in the context of our book.

A Course in Operator Theory

A Course in Operator Theory
Author: John B. Conway
Publisher: American Mathematical Soc.
Total Pages: 390
Release: 2000
Genre: Mathematics
ISBN: 0821820656

Operator theory is a significant part of many important areas of modern mathematics: functional analysis, differential equations, index theory, representation theory, mathematical physics, and more. This text covers the central themes of operator theory, presented with the excellent clarity and style that readers have come to associate with Conway's writing. Early chapters introduce and review material on $C^*$-algebras, normal operators, compact operators, and non-normal operators. Some of the major topics covered are the spectral theorem, the functional calculus, and the Fredholm index. In addition, some deep connections between operator theory and analytic functions are presented. Later chapters cover more advanced topics, such as representations of $C^*$-algebras, compact perturbations, and von Neumann algebras. Major results, such as the Sz.-Nagy Dilation Theorem, the Weyl-von Neumann-Berg Theorem, and the classification of von Neumann algebras, are covered, as is a treatment of Fredholm theory. The last chapter gives an introduction to reflexive subspaces, which along with hyperreflexive spaces, are one of the more successful episodes in the modern study of asymmetric algebras. Professor Conway's authoritative treatment makes this a compelling and rigorous course text, suitable for graduate students who have had a standard course in functional analysis.