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Author : Edward G. Effros
Publisher : American Mathematical Soc.
Page : 74 pages
File Size : 49,7 Mb
Release : 1981
Category : Mathematics
ISBN : 9780821816974
This book is largely devoted to the most elementary algebras and $C^*$-algebras, namely those which are direct limits of complex semisimple algebras.
Author : Nathanial Patrick Brown,Narutaka Ozawa
Publisher : American Mathematical Soc.
Page : 530 pages
File Size : 47,6 Mb
Release : 2008
Category : C*-algebras
ISBN : 9780821843819
$\textrm{C}*$-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.
Author : K. R. Goodearl,David Handelman
Publisher : American Mathematical Soc.
Page : 147 pages
File Size : 52,5 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 : Joan Bosa,Nathanial P. Brown,Yasuhiko Sato,Aaron Tikuisis,Stuart White,Wilhelm Winter
Publisher : American Mathematical Soc.
Page : 97 pages
File Size : 40,6 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 : Kenneth R. Davidson
Publisher : American Mathematical Society, Fields Institute
Page : 325 pages
File Size : 45,8 Mb
Release : 2023-10-04
Category : Mathematics
ISBN : 9781470475086
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.
Author : Kenneth R. Davidson
Publisher : American Mathematical Soc.
Page : 326 pages
File Size : 45,8 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 : Aidan Sims,Gábor Szabó,Dana Williams
Publisher : Springer Nature
Page : 163 pages
File Size : 48,7 Mb
Release : 2020-06-22
Category : Mathematics
ISBN : 9783030397135
This book collects the notes of the lectures given at the Advanced Course on Crossed Products, Groupoids, and Rokhlin dimension, that took place at the Centre de Recerca Matemàtica (CRM) from March 13 to March 17, 2017. The notes consist of three series of lectures. The first one was given by Dana Williams (Dartmouth College), and served as an introduction to crossed products of C*-algebras and the study of their structure. The second series of lectures was delivered by Aidan Sims (Wollongong), who gave an overview of the theory of topological groupoids (as a model for groups and group actions) and groupoid C*-algebras, with particular emphasis on the case of étale groupoids. Finally, the last series was delivered by Gábor Szabó (Copenhagen), and consisted of an introduction to Rokhlin type properties (mostly centered around the work of Hirshberg, Winter and Zacharias) with hints to the more advanced theory related to groupoids.
Author : M. Rørdam,Flemming Larsen,N. Laustsen
Publisher : Cambridge University Press
Page : 260 pages
File Size : 54,9 Mb
Release : 2000-07-20
Category : Mathematics
ISBN : 0521789443
This book provides a very elementary introduction to K-theory for C*-algebras, and is ideal for beginning graduate students.
Author : M. Rordam,E. Stormer
Publisher : Springer Science & Business Media
Page : 206 pages
File Size : 41,8 Mb
Release : 2013-04-18
Category : Mathematics
ISBN : 9783662048252
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.
Author : V. Dlab,Leonard Scott
Publisher : Springer Science & Business Media
Page : 392 pages
File Size : 40,8 Mb
Release : 2013-04-18
Category : Mathematics
ISBN : 9789401715560
Based on invited lectures at the 1992 Canadian Algebra Seminar, this volume represents an up-to-date and unique report on finite-dimensional algebras as a subject with many serious interactions with other mathematical disciplines, including algebraic groups and Lie theory, automorphic forms, sheaf theory, finite groups, and homological algebra. It will interest mathematicians and graduate students in these and related subjects as an introduction to research in an area of increasing relevance and importance.
Author : Gerald J. Murphy
Publisher : Academic Press
Page : 297 pages
File Size : 51,5 Mb
Release : 2014-06-28
Category : Mathematics
ISBN : 9780080924960
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.
Author : Jacques Dixmier
Publisher : Unknown
Page : 540 pages
File Size : 52,9 Mb
Release : 1982
Category : C*-algebras
ISBN : STANFORD:36105033046231
Almost four-fifths of this book deals with the study of C*-algebras, and the main results due, among others, to Fell, Glimm, Kadison, Kaplansky, Mackey and Segal are expounded. Because of the amount of material accumulated on unitary representations of groups, the latter pages of the book are devoted to a brief account of some aspects of this subject, particularly since the theory of groups provides some of the most interesting examples of C*-algebras. The theory of C*-algebras is still expanding rapidly, but this work remains a clear and accessible introduction to the fundamentals of the subject.
Author : Nathanial Patrick Brown
Publisher : American Mathematical Soc.
Page : 122 pages
File Size : 40,6 Mb
Release : 2006
Category : C*-algebras
ISBN : 9780821839164
Various subsets of the tracial state space of a unital C$*$-algebra are studied. The largest of these subsets has a natural interpretation as the space of invariant means. II$ 1$-factor representations of a class of C$*$-algebras considered by Sorin Popa are also studied. These algebras are shown to have an unexpected variety of II$ 1$-factor representations. In addition to developing some general theory we also show that these ideas are related to numerous other problems inoperator algebras.
Author : Karl Heinrich Hofmann,Francisco Javier Thayer
Publisher : Unknown
Page : 68 pages
File Size : 50,9 Mb
Release : 1980
Category : C*-algebras
ISBN : STANFORD:36105031602589
Author : Thomas Karl Müller
Publisher : Unknown
Page : 100 pages
File Size : 52,6 Mb
Release : 1990
Category : C*-algebras
ISBN : UCR:31210012616494
