Download or read book An Introduction to the Classification of Amenable C algebras written by Huaxin Lin and published by World Scientific. This book was released on 2001 with total page 336 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory and applications of C Oeu -algebras are related to fields ranging from operator theory, group representations and quantum mechanics, to non-commutative geometry and dynamical systems. By Gelfand transformation, the theory of C Oeu -algebras is also regarded as non-commutative topology. About a decade ago, George A. Elliott initiated the program of classification of C Oeu -algebras (up to isomorphism) by their K -theoretical data. It started with the classification of AT -algebras with real rank zero. Since then great efforts have been made to classify amenable C Oeu -algebras, a class of C Oeu -algebras that arises most naturally. For example, a large class of simple amenable C Oeu -algebras is discovered to be classifiable. The application of these results to dynamical systems has been established. This book introduces the recent development of the theory of the classification of amenable C Oeu -algebras OCo the first such attempt. The first three chapters present the basics of the theory of C Oeu -algebras which are particularly important to the theory of the classification of amenable C Oeu -algebras. Chapter 4 otters the classification of the so-called AT -algebras of real rank zero. The first four chapters are self-contained, and can serve as a text for a graduate course on C Oeu -algebras. The last two chapters contain more advanced material. In particular, they deal with the classification theorem for simple AH -algebras with real rank zero, the work of Elliott and Gong. The book contains many new proofs and some original results related to the classification of amenable C Oeu -algebras. Besides being as an introduction to the theory of the classification of amenable C Oeu -algebras, it is a comprehensive reference for those more familiar with the subject. Sample Chapter(s). Chapter 1.1: Banach algebras (260 KB). Chapter 1.2: C*-algebras (210 KB). Chapter 1.3: Commutative C*-algebras (212 KB). Chapter 1.4: Positive cones (207 KB). Chapter 1.5: Approximate identities, hereditary C*-subalgebras and quotients (230 KB). Chapter 1.6: Positive linear functionals and a Gelfand-Naimark theorem (235 KB). Chapter 1.7: Von Neumann algebras (234 KB). Chapter 1.8: Enveloping von Neumann algebras and the spectral theorem (217 KB). Chapter 1.9: Examples of C*-algebras (270 KB). Chapter 1.10: Inductive limits of C*-algebras (252 KB). Chapter 1.11: Exercises (220 KB). Chapter 1.12: Addenda (168 KB). Contents: The Basics of C Oeu -Algebras; Amenable C Oeu -Algebras and K -Theory; AF- Algebras and Ranks of C Oeu -Algebras; Classification of Simple AT -Algebras; C Oeu -Algebra Extensions; Classification of Simple Amenable C Oeu -Algebras. Readership: Researchers and graduate students in operator algebras."
Download or read book Classification of Simple C algebras Inductive Limits of Matrix Algebras over Trees written by Liangqing Li and published by American Mathematical Soc.. This book was released on 1997 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, it is shown that the simple unital C*-algebras arising as inductive limits of sequences of finite direct sums of matrix algebras over [italic capital]C([italic capital]X[subscript italic]i), where [italic capital]X[subscript italic]i are arbitrary variable trees, are classified by K-theoretical and tracial data. This result generalizes the result of George Elliott of the case of [italic capital]X[subscript italic]i = [0, 1]. The added generality is useful in the classification of more general inductive limit C*-algebras.
Download or read book Classification of Nuclear C Algebras Entropy in Operator Algebras written by M. Rordam and published by Springer Science & Business Media. This book was released on 2013-04-18 with total page 206 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Classification of Ring and C ast Algebra Direct Limits of Finite Dimensional Semisimple Real Algebras written by K. R. Goodearl and published by American Mathematical Soc.. This book was released on 1987 with total page 161 pages. Available in PDF, EPUB and Kindle. Book excerpt: Motivated by (i) Elliott's classification of direct limits of countable sequences of finite-dimensional semisimple complex algebras and complex AF C*-algebras, (ii) classical results classifying involutions on finite-dimensional semisimple complex algebras, and (iii) the classification by Handelman and Rossmann of automorphisms of period two on the algebras appearing in (i) we study the real algebras described above and completely classify them, up to isomorphism, Morita equivalence, or stable isomorphism. We also show how our classification easily distinguishes various types of algebras within the given classes, and we partially solve the problem of determining exactly which values are attained by the invariants used in classifying these algebras.
Download or read book From the Basic Homotopy Lemma to the Classification of C algebras written by Huaxin Lin and published by American Mathematical Soc.. This book was released on 2017-08-11 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book examines some recent developments in the theory of -algebras, which are algebras of operators on Hilbert spaces. An elementary introduction to the technical part of the theory is given via a basic homotopy lemma concerning a pair of almost commuting unitaries. The book presents an outline of the background as well as some recent results of the classification of simple amenable -algebras, otherwise known as the Elliott program. This includes some stable uniqueness theorems and a revisiting of Bott maps via stable homotopy. Furthermore, -theory related rotation maps are introduced. The book is based on lecture notes from the CBMS lecture sequence at the University of Wyoming in the summer of 2015.
Download or read book C Algebras by Example written by Kenneth R. Davidson and published by American Mathematical Society, Fields Institute. This book was released on 2023-10-04 with total page 325 pages. Available in PDF, EPUB and Kindle. Book excerpt: The subject of C*-algebras received a dramatic revitalization in the 1970s by the introduction of topological methods through the work of Brown, Douglas, and Fillmore on extensions of C*-algebras and Elliott's use of $K$-theory to provide a useful classification of AF algebras. These results were the beginning of a marvelous new set of tools for analyzing concrete C*-algebras. This book is an introductory graduate level text which presents the basics of the subject through a detailed analysis of several important classes of C*-algebras. The development of operator algebras in the last twenty years has been based on a careful study of these special classes. While there are many books on C*-algebras and operator algebras available, this is the first one to attempt to explain the real examples that researchers use to test their hypotheses. Topics include AF algebras, Bunce–Deddens and Cuntz algebras, the Toeplitz algebra, irrational rotation algebras, group C*-algebras, discrete crossed products, abelian C*-algebras (spectral theory and approximate unitary equivalence) and extensions. It also introduces many modern concepts and results in the subject such as real rank zero algebras, topological stable rank, quasidiagonality, and various new constructions. These notes were compiled during the author's participation in the special year on C*-algebras at The Fields Institute for Research in Mathematical Sciences during the 1994–1995 academic year. The field of C*-algebras touches upon many other areas of mathematics such as group representations, dynamical systems, physics, $K$-theory, and topology. The variety of examples offered in this text expose the student to many of these connections. Graduate students with a solid course in functional analysis should be able to read this book. This should prepare them to read much of the current literature. This book is reasonably self-contained, and the author has provided results from other areas when necessary.
Download or read book Classification of mathcal O infty Stable C Algebras written by James Gabe and published by American Mathematical Society. This book was released on 2024-02-01 with total page 128 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.
Download or read book An Introduction to K Theory for C Algebras written by M. Rørdam and published by Cambridge University Press. This book was released on 2000-07-20 with total page 260 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a very elementary introduction to K-theory for C*-algebras, and is ideal for beginning graduate students.
Download or read book textrm C Algebras and Finite Dimensional Approximations written by Nathanial Patrick Brown and published by American Mathematical Soc.. This book was released on 2008 with total page 530 pages. Available in PDF, EPUB and Kindle. Book excerpt: $\textrm{C}*$-approximation theory has provided the foundation for many of the most important conceptual breakthroughs and applications of operator algebras. This book systematically studies (most of) the numerous types of approximation properties that have been important in recent years: nuclearity, exactness, quasidiagonality, local reflexivity, and others. Moreover, it contains user-friendly proofs, insofar as that is possible, of many fundamental results that were previously quite hard to extract from the literature. Indeed, perhaps the most important novelty of the first ten chapters is an earnest attempt to explain some fundamental, but difficult and technical, results as painlessly as possible. The latter half of the book presents related topics and applications--written with researchers and advanced, well-trained students in mind. The authors have tried to meet the needs both of students wishing to learn the basics of an important area of research as well as researchers who desire a fairly comprehensive reference for the theory and applications of $\textrm{C}*$-approximation theory.
Download or read book Covering Dimension of C Algebras and 2 Coloured Classification written by Joan Bosa and published by American Mathematical Soc.. This book was released on 2019-02-21 with total page 97 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors introduce the concept of finitely coloured equivalence for unital -homomorphisms between -algebras, for which unitary equivalence is the -coloured case. They use this notion to classify -homomorphisms from separable, unital, nuclear -algebras into ultrapowers of simple, unital, nuclear, -stable -algebras with compact extremal trace space up to -coloured equivalence by their behaviour on traces; this is based on a -coloured classification theorem for certain order zero maps, also in terms of tracial data. As an application the authors calculate the nuclear dimension of non-AF, simple, separable, unital, nuclear, -stable -algebras with compact extremal trace space: it is 1. In the case that the extremal trace space also has finite topological covering dimension, this confirms the remaining open implication of the Toms-Winter conjecture. Inspired by homotopy-rigidity theorems in geometry and topology, the authors derive a “homotopy equivalence implies isomorphism” result for large classes of -algebras with finite nuclear dimension.
Download or read book Approximate Homotopy of Homomorphisms from C X into a Simple C Algebra written by Huaxin Lin and published by American Mathematical Soc.. This book was released on 2010 with total page 144 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Volume 205, number 963 (second of 5 numbers)."
Download or read book Dimensions and C ast Algebras written by Edward G. Effros and published by American Mathematical Soc.. This book was released on 1981 with total page 90 pages. Available in PDF, EPUB and Kindle. Book excerpt: Discusses elementary algebras and $C DEGREES*$-algebras, namely those which are direct limits of complex semi simple al
Download or read book Crossed Products of C Algebras Topological Dynamics and Classification written by Thierry Giordano and published by Springer. This book was released on 2018-08-28 with total page 498 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book collects the notes of the lectures given at an Advanced Course on Dynamical Systems at the Centre de Recerca Matemàtica (CRM) in Barcelona. The notes consist of four series of lectures. The first one, given by Andrew Toms, presents the basic properties of the Cuntz semigroup and its role in the classification program of simple, nuclear, separable C*-algebras. The second series of lectures, delivered by N. Christopher Phillips, serves as an introduction to group actions on C*-algebras and their crossed products, with emphasis on the simple case and when the crossed products are classifiable. The third one, given by David Kerr, treats various developments related to measure-theoretic and topological aspects of crossed products, focusing on internal and external approximation concepts, both for groups and C*-algebras. Finally, the last series of lectures, delivered by Thierry Giordano, is devoted to the theory of topological orbit equivalence, with particular attention to the classification of minimal actions by finitely generated abelian groups on the Cantor set.
Download or read book Operator Algebras and Applications written by Toke M. Carlsen and published by Springer. This book was released on 2016-07-30 with total page 348 pages. Available in PDF, EPUB and Kindle. Book excerpt: Like the first Abel Symposium, held in 2004, the Abel Symposium 2015 focused on operator algebras. It is interesting to see the remarkable advances that have been made in operator algebras over these years, which strikingly illustrate the vitality of the field. A total of 26 talks were given at the symposium on a variety of themes, all highlighting the richness of the subject. The field of operator algebras was created in the 1930s and was motivated by problems of quantum mechanics. It has subsequently developed well beyond its initial intended realm of applications and expanded into such diverse areas of mathematics as representation theory, dynamical systems, differential geometry, number theory and quantum algebra. One branch, known as “noncommutative geometry”, has become a powerful tool for studying phenomena that are beyond the reach of classical analysis. This volume includes research papers that present new results, surveys that discuss the development of a specific line of research, and articles that offer a combination of survey and research. These contributions provide a multifaceted portrait of beautiful mathematics that both newcomers to the field of operator algebras and seasoned researchers alike will appreciate.
Download or read book The Classification and Structure of C Algebra Bundles written by Maurice J. Dupré and published by American Mathematical Soc.. This book was released on 1979 with total page 91 pages. Available in PDF, EPUB and Kindle. Book excerpt: The objects of study in this paper are certain fibre spaces which arise naturally in the representation theory of C*-algebras and locally compact groups. These are a type of Banach bundle, all of whose fibres are C*-algebras. The main aim of this paper is to give a pasting homotopy type classification theory for certain classes of C*-bundles having primarily finite-dimensional fibres and thus classifying the resulting second-order bundles.
Download or read book Operator Theory Operator Algebras and Their Interactions with Geometry and Topology written by Raul E Curto and published by Springer Nature. This book was released on 2020-12-12 with total page 531 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is the proceeding of the International Workshop on Operator Theory and Applications (IWOTA) held in July 2018 in Shanghai, China. It consists of original papers, surveys and expository articles in the broad areas of operator theory, operator algebras and noncommutative topology. Its goal is to give graduate students and researchers a relatively comprehensive overview of the current status of research in the relevant fields. The book is also a special volume dedicated to the memory of Ronald G. Douglas who passed away on February 27, 2018 at the age of 79. Many of the contributors are Douglas’ students and past collaborators. Their articles attest and commemorate his life-long contribution and influence to these fields.
Download or read book Classification of Direct Limits of Even Cuntz circle Algebras written by Huaxin Lin and published by American Mathematical Soc.. This book was released on 1995 with total page 132 pages. Available in PDF, EPUB and Kindle. Book excerpt: does not need NBB copy
