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| Author | : James Gabe |
| Publisher | : American Mathematical Society |
| Total Pages | : 128 |
| Release | : 2024-02-01 |
| Genre | : Mathematics |
| ISBN | : 1470467933 |
View the abstract.
| Author | : Huaxin Lin |
| Publisher | : World Scientific |
| Total Pages | : 336 |
| Release | : 2001 |
| Genre | : Mathematics |
| ISBN | : 9789812799883 |
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."
| Author | : Joan Bosa |
| Publisher | : |
| Total Pages | : |
| Release | : 2019 |
| Genre | : C*-algebras |
| ISBN | : 9781470449490 |
| Author | : K. R. Goodearl |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 161 |
| Release | : 1987 |
| Genre | : Mathematics |
| ISBN | : 082182435X |
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.
| Author | : Kenneth R. Davidson |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 326 |
| Release | : 1996 |
| Genre | : Mathematics |
| ISBN | : 0821805991 |
An introductory graduate level text presenting the basics of the subject through a detailed analysis of several important classes of C*-algebras, those which are the basis of the development of operator algebras. Explains the real examples that researchers use to test their hypotheses, and introduces modern concepts and results such as real rank zero algebras, topological stable rank, and quasidiagonality. Includes chapter exercises with hints. For graduate students with a foundation in functional analysis. Annotation copyright by Book News, Inc., Portland, OR
| Author | : Karen R. Strung |
| Publisher | : Springer Nature |
| Total Pages | : 322 |
| Release | : 2020-12-15 |
| Genre | : Mathematics |
| ISBN | : 3030474658 |
This book is directed towards graduate students that wish to start from the basic theory of C*-algebras and advance to an overview of some of the most spectacular results concerning the structure of nuclear C*-algebras. The text is divided into three parts. First, elementary notions, classical theorems and constructions are developed. Then, essential examples in the theory, such as crossed products and the class of quasidiagonal C*-algebras, are examined, and finally, the Elliott invariant, the Cuntz semigroup, and the Jiang-Su algebra are defined. It is shown how these objects have played a fundamental role in understanding the fine structure of nuclear C*-algebras. To help understanding the theory, plenty of examples, treated in detail, are included. This volume will also be valuable to researchers in the area as a reference guide. It contains an extensive reference list to guide readers that wish to travel further.
| Author | : Liangqing Li |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 140 |
| Release | : 1997-01-01 |
| Genre | : Mathematics |
| ISBN | : 9780821863282 |
In this book, it is shown that the simple unital C-]* algebras arising as inductive limits of sequences of finite direct sums of matrix algebras over C(X[i), where X[i are arbitrary variable trees, are classified by K-theoretical and tracial data. This result generalizes the result of George Elliott of the case X[i = [0, 1]. The added generality is useful in the classification of more general inductive limit C]*-algebras.
| Author | : Mikael Rørdam |
| Publisher | : |
| Total Pages | : 198 |
| Release | : 2002 |
| Genre | : C*-algebras |
| ISBN | : |
| Author | : John Voight |
| Publisher | : Springer Nature |
| Total Pages | : 877 |
| Release | : 2021-06-28 |
| Genre | : Mathematics |
| ISBN | : 3030566943 |
This open access textbook presents a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, a subject with applications in diverse areas of mathematics. Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature. Numerous pathways offer explorations in many different directions, while the unified treatment makes this book an essential reference for students and researchers alike. Divided into five parts, the book begins with a basic introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields, including the relationship to quadratic forms. An in-depth exploration of the arithmetic of quaternion algebras and orders follows. The third part considers analytic aspects, starting with zeta functions and then passing to an idelic approach, offering a pathway from local to global that includes strong approximation. Applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology follow, relating geometric and topological properties to arithmetic invariants. Arithmetic geometry completes the volume, including quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces. Quaternion Algebras encompasses a vast wealth of knowledge at the intersection of many fields. Graduate students interested in algebra, geometry, and number theory will appreciate the many avenues and connections to be explored. Instructors will find numerous options for constructing introductory and advanced courses, while researchers will value the all-embracing treatment. Readers are assumed to have some familiarity with algebraic number theory and commutative algebra, as well as the fundamentals of linear algebra, topology, and complex analysis. More advanced topics call upon additional background, as noted, though essential concepts and motivation are recapped throughout.
| Author | : Benoit Fresse |
| Publisher | : Springer |
| Total Pages | : 304 |
| Release | : 2009-04-20 |
| Genre | : Mathematics |
| ISBN | : 3540890564 |
This monograph presents a review of the basis of operad theory. It also studies structures of modules over operads as a new device to model functors between categories of algebras as effectively as operads model categories of algebras.
