K-Theory for Group C*-Algebras and Semigroup C*-Algebras PDF Download
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Author: Joachim Cuntz
Publisher: Birkhäuser
ISBN: 3319599151
Category : Mathematics
Languages : en
Pages : 325
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Book Description
This book gives an account of the necessary background for group algebras and crossed products for actions of a group or a semigroup on a space and reports on some very recently developed techniques with applications to particular examples. Much of the material is available here for the first time in book form. The topics discussed are among the most classical and intensely studied C*-algebras. They are important for applications in fields as diverse as the theory of unitary group representations, index theory, the topology of manifolds or ergodic theory of group actions. Part of the most basic structural information for such a C*-algebra is contained in its K-theory. The determination of the K-groups of C*-algebras constructed from group or semigroup actions is a particularly challenging problem. Paul Baum and Alain Connes proposed a formula for the K-theory of the reduced crossed product for a group action that would permit, in principle, its computation. By work of many hands, the formula has by now been verified for very large classes of groups and this work has led to the development of a host of new techniques. An important ingredient is Kasparov's bivariant K-theory. More recently, also the C*-algebras generated by the regular representation of a semigroup as well as the crossed products for actions of semigroups by endomorphisms have been studied in more detail. Intriguing examples of actions of such semigroups come from ergodic theory as well as from algebraic number theory. The computation of the K-theory of the corresponding crossed products needs new techniques. In cases of interest the K-theory of the algebras reflects ergodic theoretic or number theoretic properties of the action.
Author: Joachim Cuntz
Publisher: Birkhäuser
ISBN: 3319599151
Category : Mathematics
Languages : en
Pages : 325
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Book Description
This book gives an account of the necessary background for group algebras and crossed products for actions of a group or a semigroup on a space and reports on some very recently developed techniques with applications to particular examples. Much of the material is available here for the first time in book form. The topics discussed are among the most classical and intensely studied C*-algebras. They are important for applications in fields as diverse as the theory of unitary group representations, index theory, the topology of manifolds or ergodic theory of group actions. Part of the most basic structural information for such a C*-algebra is contained in its K-theory. The determination of the K-groups of C*-algebras constructed from group or semigroup actions is a particularly challenging problem. Paul Baum and Alain Connes proposed a formula for the K-theory of the reduced crossed product for a group action that would permit, in principle, its computation. By work of many hands, the formula has by now been verified for very large classes of groups and this work has led to the development of a host of new techniques. An important ingredient is Kasparov's bivariant K-theory. More recently, also the C*-algebras generated by the regular representation of a semigroup as well as the crossed products for actions of semigroups by endomorphisms have been studied in more detail. Intriguing examples of actions of such semigroups come from ergodic theory as well as from algebraic number theory. The computation of the K-theory of the corresponding crossed products needs new techniques. In cases of interest the K-theory of the algebras reflects ergodic theoretic or number theoretic properties of the action.
Author: M. Rørdam
Publisher: Cambridge University Press
ISBN: 9780521789448
Category : Mathematics
Languages : en
Pages : 260
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Book Description
This book provides a very elementary introduction to K-theory for C*-algebras, and is ideal for beginning graduate students.
Author: Karen R. Strung
Publisher: Springer Nature
ISBN: 3030474658
Category : Mathematics
Languages : en
Pages : 333
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Book Description
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: Gerald J. Murphy
Publisher: Academic Press
ISBN: 0080924964
Category : Mathematics
Languages : en
Pages : 297
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Book Description
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: Michael Atiyah
Publisher: CRC Press
ISBN: 0429973179
Category : Mathematics
Languages : en
Pages : 181
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Book Description
These notes are based on the course of lectures I gave at Harvard in the fall of 1964. They constitute a self-contained account of vector bundles and K-theory assuming only the rudiments of point-set topology and linear algebra. One of the features of the treatment is that no use is made of ordinary homology or cohomology theory. In fact, rational cohomology is defined in terms of K-theory.The theory is taken as far as the solution of the Hopf invariant problem and a start is mode on the J-homomorphism. In addition to the lecture notes proper, two papers of mine published since 1964 have been reproduced at the end. The first, dealing with operations, is a natural supplement to the material in Chapter III. It provides an alternative approach to operations which is less slick but more fundamental than the Grothendieck method of Chapter III, and it relates operations and filtration. Actually, the lectures deal with compact spaces, not cell-complexes, and so the skeleton-filtration does not figure in the notes. The second paper provides a new approach to K-theory and so fills an obvious gap in the lecture notes.
Author: Ruy Exel
Publisher: American Mathematical Soc.
ISBN: 1470437856
Category : Mathematics
Languages : en
Pages : 330
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Book Description
Partial dynamical systems, originally developed as a tool to study algebras of operators in Hilbert spaces, has recently become an important branch of algebra. Its most powerful results allow for understanding structural properties of algebras, both in the purely algebraic and in the C*-contexts, in terms of the dynamical properties of certain systems which are often hiding behind algebraic structures. The first indication that the study of an algebra using partial dynamical systems may be helpful is the presence of a grading. While the usual theory of graded algebras often requires gradings to be saturated, the theory of partial dynamical systems is especially well suited to treat nonsaturated graded algebras which are in fact the source of the notion of “partiality”. One of the main results of the book states that every graded algebra satisfying suitable conditions may be reconstructed from a partial dynamical system via a process called the partial crossed product. Running in parallel with partial dynamical systems, partial representations of groups are also presented and studied in depth. In addition to presenting main theoretical results, several specific examples are analyzed, including Wiener–Hopf algebras and graph C*-algebras.
Author: Alfred Hoblitzelle Clifford
Publisher: American Mathematical Soc.
ISBN: 0821802720
Category : Group theory
Languages : en
Pages : 370
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Book Description
Author: Alain Connes
Publisher: Springer
ISBN: 3540397027
Category : Mathematics
Languages : en
Pages : 364
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Book Description
Noncommutative Geometry is one of the most deep and vital research subjects of present-day Mathematics. Its development, mainly due to Alain Connes, is providing an increasing number of applications and deeper insights for instance in Foliations, K-Theory, Index Theory, Number Theory but also in Quantum Physics of elementary particles. The purpose of the Summer School in Martina Franca was to offer a fresh invitation to the subject and closely related topics; the contributions in this volume include the four main lectures, cover advanced developments and are delivered by prominent specialists.
Author: Rufus Willett
Publisher: Cambridge University Press
ISBN: 1108853110
Category : Mathematics
Languages : en
Pages : 595
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Book Description
Index theory studies the solutions to differential equations on geometric spaces, their relation to the underlying geometry and topology, and applications to physics. If the space of solutions is infinite dimensional, it becomes necessary to generalise the classical Fredholm index using tools from the K-theory of operator algebras. This leads to higher index theory, a rapidly developing subject with connections to noncommutative geometry, large-scale geometry, manifold topology and geometry, and operator algebras. Aimed at geometers, topologists and operator algebraists, this book takes a friendly and concrete approach to this exciting theory, focusing on the main conjectures in the area and their applications outside of it. A well-balanced combination of detailed introductory material (with exercises), cutting-edge developments and references to the wider literature make this a valuable guide to this active area for graduate students and experts alike.
Author: Dana P. Williams
Publisher: American Mathematical Soc.
ISBN: 0821842420
Category : Mathematics
Languages : en
Pages : 546
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Book Description
The theory of crossed products is extremely rich and intriguing. There are applications not only to operator algebras, but to subjects as varied as noncommutative geometry and mathematical physics. This book provides a detailed introduction to this vast subject suitable for graduate students and others whose research has contact with crossed product $C*$-algebras. in addition to providing the basic definitions and results, the main focus of this book is the fine ideal structure of crossed products as revealed by the study of induced representations via the Green-Mackey-Rieffel machine. in particular, there is an in-depth analysis of the imprimitivity theorems on which Rieffel's theory of induced representations and Morita equivalence of $C*$-algebras are based. There is also a detailed treatment of the generalized Effros-Hahn conjecture and its proof due to Gootman, Rosenberg, and Sauvageot. This book is meant to be self-contained and accessible to any graduate student coming out of a first course on operator algebras. There are appendices that deal with ancillary subjects, which while not central to the subject, are nevertheless crucial for a complete understanding of the material. Some of the appendices will be of independent interest. to view another book by this author, please visit Morita Equivalence and Continuous-Trace $C*$-Algebras.
