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Author : James Gabe
Publisher : American Mathematical Society
Page : 128 pages
File Size : 44,9 Mb
Release : 2024-02-01
Category : Mathematics
ISBN : 9781470467937
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Author : James Gabe
Publisher : Unknown
Page : 0 pages
File Size : 44,8 Mb
Release : 2024
Category : Operator algebras
ISBN : 1470477181
Keywords: Purely infinite C*-algebras, classification of non-simple C*-algebras, the Kirchberg-Phillips theorem.
Author : Joan Bosa,Nathanial P. Brown,Yasuhiko Sato,Aaron Tikuisis,Stuart White,Wilhelm Winter
Publisher : American Mathematical Soc.
Page : 97 pages
File Size : 48,9 Mb
Release : 2019-02-21
Category : C*-algebras
ISBN : 9781470434700
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.
Author : K. R. Goodearl,David Handelman
Publisher : American Mathematical Soc.
Page : 147 pages
File Size : 44,6 Mb
Release : 1987
Category : Mathematics
ISBN : 9780821824351
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 : Karen R. Strung
Publisher : Springer Nature
Page : 322 pages
File Size : 42,6 Mb
Release : 2020-12-15
Category : Mathematics
ISBN : 9783030474652
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 : Kenneth R. Davidson
Publisher : American Mathematical Soc.
Page : 326 pages
File Size : 54,5 Mb
Release : 1996
Category : C*-algebras
ISBN : 9780821805992
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 : Liangqing Li
Publisher : American Mathematical Soc.
Page : 140 pages
File Size : 50,5 Mb
Release : 1997-01-01
Category : Mathematics
ISBN : 0821863282
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 : John Voight
Publisher : Springer Nature
Page : 877 pages
File Size : 53,7 Mb
Release : 2021-06-28
Category : Mathematics
ISBN : 9783030566944
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 : Iain Raeburn
Publisher : American Mathematical Soc.
Page : 130 pages
File Size : 53,7 Mb
Release : 2005
Category : Mathematics
ISBN : 9780821836606
Graph algebras are a family of operator algebras which are associated to directed graphs. These algebras have an attractive structure theory in which algebraic properties of the algebra are related to the behavior of paths in the underlying graph. In the past few years there has been a great deal of activity in this area, and graph algebras have cropped up in a surprising variety of situations, including non-abelian duality, non-commutative geometry, and the classification of simple $C*$-algebras. The first part of the book provides an introduction to the subject suitable for students who have seen a first course on the basics of $C*$-algebras. In the second part, the author surveys the literature on the structure theory of graph algebras, highlights some applications of this theory, and discusses several recent generalizations which seem particularly promising. The volume is suitable for graduate students and research mathematicians interested in graph theory and operator algebras.
Author : Benoit Fresse
Publisher : Springer
Page : 304 pages
File Size : 53,8 Mb
Release : 2009-04-20
Category : Mathematics
ISBN : 9783540890560
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.
Author : David Mond,Juan J. Nuño-Ballesteros
Publisher : Springer Nature
Page : 567 pages
File Size : 47,6 Mb
Release : 2020-01-23
Category : Mathematics
ISBN : 9783030344405
The first monograph on singularities of mappings for many years, this book provides an introduction to the subject and an account of recent developments concerning the local structure of complex analytic mappings. Part I of the book develops the now classical real C∞ and complex analytic theories jointly. Standard topics such as stability, deformation theory and finite determinacy, are covered in this part. In Part II of the book, the authors focus on the complex case. The treatment is centred around the idea of the "nearby stable object" associated to an unstable map-germ, which includes in particular the images and discriminants of stable perturbations of unstable singularities. This part includes recent research results, bringing the reader up to date on the topic. By focusing on singularities of mappings, rather than spaces, this book provides a necessary addition to the literature. Many examples and exercises, as well as appendices on background material, make it an invaluable guide for graduate students and a key reference for researchers. A number of graduate level courses on singularities of mappings could be based on the material it contains.
Author : Gilles Pisier
Publisher : Cambridge University Press
Page : 495 pages
File Size : 48,5 Mb
Release : 2020-02-27
Category : Mathematics
ISBN : 9781108479011
Presents an important open problem on operator algebras in a style accessible to young researchers or Ph.D. students.
Author : Boris Zilber
Publisher : American Mathematical Soc.
Page : 132 pages
File Size : 43,6 Mb
Release : 2024-07-04
Category : Mathematics
ISBN : 0821897454
The 1970s saw the appearance and development in categoricity theory of a tendency to focus on the study and description of uncountably categorical theories in various special classes defined by natural algebraic or syntactic conditions. There have thus been studies of uncountably categorical theories of groups and rings, theories of a one-place function, universal theories of semigroups, quasivarieties categorical in infinite powers, and Horn theories. In Uncountably Categorical Theories , this research area is referred to as the special classification theory of categoricity. Zilber's goal is to develop a structural theory of categoricity, using methods and results of the special classification theory, and to construct on this basis a foundation for a general classification theory of categoricity, that is, a theory aimed at describing large classes of uncountably categorical structures not restricted by any syntactic or algebraic conditions.
Author : Ilijas Farah
Publisher : Springer Nature
Page : 517 pages
File Size : 50,9 Mb
Release : 2019-12-24
Category : Mathematics
ISBN : 9783030270933
This book explores and highlights the fertile interaction between logic and operator algebras, which in recent years has led to the resolution of several long-standing open problems on C*-algebras. The interplay between logic and operator algebras (C*-algebras, in particular) is relatively young and the author is at the forefront of this interaction. The deep level of scholarship contained in these pages is evident and opens doors to operator algebraists interested in learning about the set-theoretic methods relevant to their field, as well as to set-theorists interested in expanding their view to the non-commutative realm of operator algebras. Enough background is included from both subjects to make the book a convenient, self-contained source for students. A fair number of the exercises form an integral part of the text. They are chosen to widen and deepen the material from the corresponding chapters. Some other exercises serve as a warmup for the latter chapters.
Author : Michael A. Hill,Michael J. Hopkins,Douglas C. Ravenel
Publisher : Cambridge University Press
Page : 881 pages
File Size : 44,7 Mb
Release : 2021-07-29
Category : Mathematics
ISBN : 9781108831444
A complete and definitive account of the authors' resolution of the Kervaire invariant problem in stable homotopy theory.