Leavitt Path Algebras And Classical K Theory

Author : A. A. Ambily,Roozbeh Hazrat,B. Sury
Publisher : Springer Nature
Page : 340 pages
File Size : 47,8 Mb
Release : 2020-01-17
Category : Mathematics
ISBN : 9789811516115

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Leavitt Path Algebras and Classical K-Theory by A. A. Ambily,Roozbeh Hazrat,B. Sury Pdf

The book offers a comprehensive introduction to Leavitt path algebras (LPAs) and graph C*-algebras. Highlighting their significant connection with classical K-theory—which plays an important role in mathematics and its related emerging fields—this book allows readers from diverse mathematical backgrounds to understand and appreciate these structures. The articles on LPAs are mostly of an expository nature and the ones dealing with K-theory provide new proofs and are accessible to interested students and beginners of the field. It is a useful resource for graduate students and researchers working in this field and related areas, such as C*-algebras and symbolic dynamics.

Author : Gene Abrams,Ara, Pere,Mercedes Siles Molina
Publisher : Unknown
Page : 289 pages
File Size : 49,5 Mb
Release : 2017
Category : Algebra
ISBN : 1447173457

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Leavitt Path Algebras by Gene Abrams,Ara, Pere,Mercedes Siles Molina Pdf

Author : Joachim Cuntz,Siegfried Echterhoff,Xin Li,Guoliang Yu
Publisher : Birkhäuser
Page : 322 pages
File Size : 55,8 Mb
Release : 2017-10-24
Category : Mathematics
ISBN : 9783319599151

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K-Theory for Group C*-Algebras and Semigroup C*-Algebras by Joachim Cuntz,Siegfried Echterhoff,Xin Li,Guoliang Yu Pdf

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,Flemming Larsen,N. Laustsen
Publisher : Cambridge University Press
Page : 260 pages
File Size : 49,7 Mb
Release : 2000-07-20
Category : Mathematics
ISBN : 0521789443

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An Introduction to K-Theory for C*-Algebras by M. Rørdam,Flemming Larsen,N. Laustsen Pdf

This book provides a very elementary introduction to K-theory for C*-algebras, and is ideal for beginning graduate students.

Author : Gene Abrams,Pere Ara,Mercedes Siles Molina
Publisher : Springer
Page : 289 pages
File Size : 43,9 Mb
Release : 2017-11-30
Category : Mathematics
ISBN : 9781447173441

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Leavitt Path Algebras by Gene Abrams,Pere Ara,Mercedes Siles Molina Pdf

This book offers a comprehensive introduction by three of the leading experts in the field, collecting fundamental results and open problems in a single volume. Since Leavitt path algebras were first defined in 2005, interest in these algebras has grown substantially, with ring theorists as well as researchers working in graph C*-algebras, group theory and symbolic dynamics attracted to the topic. Providing a historical perspective on the subject, the authors review existing arguments, establish new results, and outline the major themes and ring-theoretic concepts, such as the ideal structure, Z-grading and the close link between Leavitt path algebras and graph C*-algebras. The book also presents key lines of current research, including the Algebraic Kirchberg Phillips Question, various additional classification questions, and connections to noncommutative algebraic geometry. Leavitt Path Algebras will appeal to graduate students and researchers working in the field and related areas, such as C*-algebras and symbolic dynamics. With its descriptive writing style, this book is highly accessible.

Author : Aderemi Kuku
Publisher : CRC Press
Page : 442 pages
File Size : 48,5 Mb
Release : 2016-04-19
Category : Mathematics
ISBN : 9781420011128

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Representation Theory and Higher Algebraic K-Theory by Aderemi Kuku Pdf

Representation Theory and Higher Algebraic K-Theory is the first book to present higher algebraic K-theory of orders and group rings as well as characterize higher algebraic K-theory as Mackey functors that lead to equivariant higher algebraic K-theory and their relative generalizations. Thus, this book makes computations of higher K-theory of grou

Author : R. Keith Dennis,Claudio Pedrini,Michael R. Stein
Publisher : American Mathematical Soc.
Page : 230 pages
File Size : 53,5 Mb
Release : 1992
Category : Mathematics
ISBN : 9780821851302

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Algebraic K-theory, Commutative Algebra, and Algebraic Geometry by R. Keith Dennis,Claudio Pedrini,Michael R. Stein Pdf

In the mid-1960s, several Italian mathematicians began to study the connections between classical arguments in commutative algebra and algebraic geometry, and the contemporaneous development of algebraic $K$-theory in the U.S. These connections were exemplified by the work of Andreotti-Bombieri, Salmon, and Traverso on seminormality, and by Bass-Murthy on the Picard groups of polynomial rings. Interactions proceeded far beyond this initial point to encompass Chow groups of singular varieties, complete intersections, and applications of $K$-theory to arithmetic and real geometry. This volume contains the proceedings from a U.S.-Italy Joint Summer Seminar, which focused on this circle of ideas. The conference, held in June 1989 in Santa Margherita Ligure, Italy, was supported jointly by the Consiglio Nazionale delle Ricerche and the National Science Foundation. The book contains contributions from some of the leading experts in this area.

Author : Guillermo Cortiñas
Publisher : European Mathematical Society
Page : 460 pages
File Size : 42,6 Mb
Release : 2008
Category : K-theory
ISBN : 3037190604

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K-theory and Noncommutative Geometry by Guillermo Cortiñas Pdf

Since its inception 50 years ago, K-theory has been a tool for understanding a wide-ranging family of mathematical structures and their invariants: topological spaces, rings, algebraic varieties and operator algebras are the dominant examples. The invariants range from characteristic classes in cohomology, determinants of matrices, Chow groups of varieties, as well as traces and indices of elliptic operators. Thus K-theory is notable for its connections with other branches of mathematics. Noncommutative geometry develops tools which allow one to think of noncommutative algebras in the same footing as commutative ones: as algebras of functions on (noncommutative) spaces. The algebras in question come from problems in various areas of mathematics and mathematical physics; typical examples include algebras of pseudodifferential operators, group algebras, and other algebras arising from quantum field theory. To study noncommutative geometric problems one considers invariants of the relevant noncommutative algebras. These invariants include algebraic and topological K-theory, and also cyclic homology, discovered independently by Alain Connes and Boris Tsygan, which can be regarded both as a noncommutative version of de Rham cohomology and as an additive version of K-theory. There are primary and secondary Chern characters which pass from K-theory to cyclic homology. These characters are relevant both to noncommutative and commutative problems and have applications ranging from index theorems to the detection of singularities of commutative algebraic varieties. The contributions to this volume represent this range of connections between K-theory, noncommmutative geometry, and other branches of mathematics.

Author : Paul Frank Baum,Ralf Meyer,Rubén Sánchez-García,Marco Schlichting,Bertrand Toën
Publisher : Springer Science & Business Media
Page : 322 pages
File Size : 51,5 Mb
Release : 2010-11-05
Category : Mathematics
ISBN : 9783642157073

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Topics in Algebraic and Topological K-Theory by Paul Frank Baum,Ralf Meyer,Rubén Sánchez-García,Marco Schlichting,Bertrand Toën Pdf

This volume is an introductory textbook to K-theory, both algebraic and topological, and to various current research topics within the field, including Kasparov's bivariant K-theory, the Baum-Connes conjecture, the comparison between algebraic and topological K-theory of topological algebras, the K-theory of schemes, and the theory of dg-categories.

Author : Joachim Cuntz,Jonathan Rosenberg
Publisher : Springer Science & Business Media
Page : 268 pages
File Size : 40,5 Mb
Release : 2007-07-19
Category : Mathematics
ISBN : 9783764383985

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Topological and Bivariant K-Theory by Joachim Cuntz,Jonathan Rosenberg Pdf

Topological K-theory is one of the most important invariants for noncommutative algebras. Bott periodicity, homotopy invariance, and various long exact sequences distinguish it from algebraic K-theory. This book describes a bivariant K-theory for bornological algebras, which provides a vast generalization of topological K-theory. In addition, it details other approaches to bivariant K-theories for operator algebras. The book studies a number of applications, including K-theory of crossed products, the Baum-Connes assembly map, twisted K-theory with some of its applications, and some variants of the Atiyah-Singer Index Theorem.

Author : Toke M. Carlsen,Nadia S. Larsen,Sergey Neshveyev,Christian Skau
Publisher : Springer
Page : 348 pages
File Size : 49,7 Mb
Release : 2016-07-30
Category : Mathematics
ISBN : 9783319392868

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Operator Algebras and Applications by Toke M. Carlsen,Nadia S. Larsen,Sergey Neshveyev,Christian Skau Pdf

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.

Author : Bruce Blackadar
Publisher : Springer Science & Business Media
Page : 347 pages
File Size : 46,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461395720

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K-Theory for Operator Algebras by Bruce Blackadar Pdf

K -Theory has revolutionized the study of operator algebras in the last few years. As the primary component of the subject of "noncommutative topol ogy," K -theory has opened vast new vistas within the structure theory of C* algebras, as well as leading to profound and unexpected applications of opera tor algebras to problems in geometry and topology. As a result, many topolo gists and operator algebraists have feverishly begun trying to learn each others' subjects, and it appears certain that these two branches of mathematics have become deeply and permanently intertwined. Despite the fact that the whole subject is only about a decade old, operator K -theory has now reached a state of relative stability. While there will undoubtedly be many more revolutionary developments and applications in the future, it appears the basic theory has more or less reached a "final form." But because of the newness of the theory, there has so far been no comprehensive treatment of the subject. It is the ambitious goal of these notes to fill this gap. We will develop the K -theory of Banach algebras, the theory of extensions of C*-algebras, and the operator K -theory of Kasparov from scratch to its most advanced aspects. We will not treat applications in detail; however, we will outline the most striking of the applications to date in a section at the end, as well as mentioning others at suitable points in the text.

Author : William Browder
Publisher : Princeton University Press
Page : 567 pages
File Size : 53,9 Mb
Release : 2016-03-02
Category : Mathematics
ISBN : 9781400882113

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Algebraic Topology and Algebraic K-Theory (AM-113), Volume 113 by William Browder Pdf

This book contains accounts of talks held at a symposium in honor of John C. Moore in October 1983 at Princeton University, The work includes papers in classical homotopy theory, homological algebra, rational homotopy theory, algebraic K-theory of spaces, and other subjects.

Author : Bjørn Ian Dundas,Thomas G. Goodwillie,Randy McCarthy
Publisher : Springer Science & Business Media
Page : 447 pages
File Size : 44,9 Mb
Release : 2012-09-06
Category : Mathematics
ISBN : 9781447143932

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The Local Structure of Algebraic K-Theory by Bjørn Ian Dundas,Thomas G. Goodwillie,Randy McCarthy Pdf

Algebraic K-theory encodes important invariants for several mathematical disciplines, spanning from geometric topology and functional analysis to number theory and algebraic geometry. As is commonly encountered, this powerful mathematical object is very hard to calculate. Apart from Quillen's calculations of finite fields and Suslin's calculation of algebraically closed fields, few complete calculations were available before the discovery of homological invariants offered by motivic cohomology and topological cyclic homology. This book covers the connection between algebraic K-theory and Bökstedt, Hsiang and Madsen's topological cyclic homology and proves that the difference between the theories are ‘locally constant’. The usefulness of this theorem stems from being more accessible for calculations than K-theory, and hence a single calculation of K-theory can be used with homological calculations to obtain a host of ‘nearby’ calculations in K-theory. For instance, Quillen's calculation of the K-theory of finite fields gives rise to Hesselholt and Madsen's calculations for local fields, and Voevodsky's calculations for the integers give insight into the diffeomorphisms of manifolds. In addition to the proof of the full integral version of the local correspondence between K-theory and topological cyclic homology, the book provides an introduction to the necessary background in algebraic K-theory and highly structured homotopy theory; collecting all necessary tools into one common framework. It relies on simplicial techniques, and contains an appendix summarizing the methods widely used in the field. The book is intended for graduate students and scientists interested in algebraic K-theory, and presupposes a basic knowledge of algebraic topology.

Author : Vasudevan Srinivas
Publisher : Springer Science & Business Media
Page : 357 pages
File Size : 49,9 Mb
Release : 2009-05-21
Category : Mathematics
ISBN : 9780817647391

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Algebraic K-Theory by Vasudevan Srinivas Pdf

Algebraic K-Theory has become an increasingly active area of research. With its connections to algebra, algebraic geometry, topology, and number theory, it has implications for a wide variety of researchers and students in mathematics. This book is based on lectures given by the author at the Tata Institute in Bombay and elsewhere. This new edition includes an appendix on algebraic geometry that contains required definitions and results needed to understand the core of the book.