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Author: Christian Groenbaek Publisher: CRC Press ISBN: 9781584880103 Category : Mathematics Languages : en Pages : 126
Book Description
Recent work by Cuntz and Quillen on bivariant periodic cyclic homology has caused quite a revolution in the subject. In this self-contained exposition, the author's purpose is to understand the functorial properties of the Cuntz-Quillen theory, which motivaties his explorations of what he calls cyclic pro-homology. Simply stated, the cyclic pro-homology of an (associative) algebra A is short for the Z/2 Z-graded inverse system of cyclic homology groups of A, considered as a pro-vector space. The author finds that this functor, taking algebras over a field k of characteristic zero into the category of pro-k-vector spaces, is remarkable. He presents a proof that it is excisive and that it satisfies a Künneth isomorphism for the tensor product of algebras. He explains the relation to the Cuntz-Quillen groups in a Universal Coefficient Theorem and in a Milnor lim1-sequence. This enables the lifting - to some extent- of the nice properties of cyclic pro-homology properties to the Cuntz Quillen theory itself. It is interesting to note that for the excision result, this lifting procedure goes through without constraints. For those new to cyclic homology, Dr. Grønbaek takes care to provide an introduction to the subject, including the motivation for the theory, definitions, and fundamental results, and establishes the homological machinery needed for application to the Cuntz-Quillen theory. Mathematicians interested in cyclic homology-especially ring theorists using homological methods-will find this work original, enlightening, and thought-provoking. The author leaves the door open for deeper study into excision for the Cuntz-Quillen theory for a class of topological algebras, such as the category of m-algebras considered by Cuntz.
Author: Christian Groenbaek Publisher: CRC Press ISBN: 9781584880103 Category : Mathematics Languages : en Pages : 126
Book Description
Recent work by Cuntz and Quillen on bivariant periodic cyclic homology has caused quite a revolution in the subject. In this self-contained exposition, the author's purpose is to understand the functorial properties of the Cuntz-Quillen theory, which motivaties his explorations of what he calls cyclic pro-homology. Simply stated, the cyclic pro-homology of an (associative) algebra A is short for the Z/2 Z-graded inverse system of cyclic homology groups of A, considered as a pro-vector space. The author finds that this functor, taking algebras over a field k of characteristic zero into the category of pro-k-vector spaces, is remarkable. He presents a proof that it is excisive and that it satisfies a Künneth isomorphism for the tensor product of algebras. He explains the relation to the Cuntz-Quillen groups in a Universal Coefficient Theorem and in a Milnor lim1-sequence. This enables the lifting - to some extent- of the nice properties of cyclic pro-homology properties to the Cuntz Quillen theory itself. It is interesting to note that for the excision result, this lifting procedure goes through without constraints. For those new to cyclic homology, Dr. Grønbaek takes care to provide an introduction to the subject, including the motivation for the theory, definitions, and fundamental results, and establishes the homological machinery needed for application to the Cuntz-Quillen theory. Mathematicians interested in cyclic homology-especially ring theorists using homological methods-will find this work original, enlightening, and thought-provoking. The author leaves the door open for deeper study into excision for the Cuntz-Quillen theory for a class of topological algebras, such as the category of m-algebras considered by Cuntz.
Author: Christian Groenbaek Publisher: ISBN: 9780582368941 Category : Languages : en Pages : 104
Book Description
Recent work by Cuntz and Quillen on bivariant periodic cyclic homology has caused quite a revolution in the subject. In this self-contained exposition, the author's purpose is to understand the functorial prope rties of the Cuntz-Quillen theory, which motivaties his explorations o f what he calls cyclic pro-homology.
Author: Joachim Cuntz Publisher: Springer Science & Business Media ISBN: 9783540404699 Category : Mathematics Languages : en Pages : 160
Book Description
Contributions by three authors treat aspects of noncommutative geometry that are related to cyclic homology. The authors give rather complete accounts of cyclic theory from different points of view. The connections between (bivariant) K-theory and cyclic theory via generalized Chern-characters are discussed in detail. Cyclic theory is the natural setting for a variety of general abstract index theorems. A survey of such index theorems is given and the concepts and ideas involved in these theorems are explained.
Author: Ralf Meyer Publisher: European Mathematical Society ISBN: 9783037190395 Category : Mathematics Languages : en Pages : 376
Book Description
Periodic cyclic homology is a homology theory for non-commutative algebras that plays a similar role in non-commutative geometry as de Rham cohomology for smooth manifolds. While it produces good results for algebras of smooth or polynomial functions, it fails for bigger algebras such as most Banach algebras or C*-algebras. Analytic and local cyclic homology are variants of periodic cyclic homology that work better for such algebras. In this book, the author develops and compares these theories, emphasizing their homological properties. This includes the excision theorem, invariance under passage to certain dense subalgebras, a Universal Coefficient Theorem that relates them to $K$-theory, and the Chern-Connes character for $K$-theory and $K$-homology. The cyclic homology theories studied in this text require a good deal of functional analysis in bornological vector spaces, which is supplied in the first chapters. The focal points here are the relationship with inductive systems and the functional calculus in non-commutative bornological algebras. Some chapters are more elementary and independent of the rest of the book and will be of interest to researchers and students working on functional analysis and its applications.
Author: Joachim Cuntz Publisher: Springer Science & Business Media ISBN: 3662064448 Category : Mathematics Languages : en Pages : 147
Book Description
Contributions by three authors treat aspects of noncommutative geometry that are related to cyclic homology. The authors give rather complete accounts of cyclic theory from different points of view. The connections between (bivariant) K-theory and cyclic theory via generalized Chern-characters are discussed in detail. Cyclic theory is the natural setting for a variety of general abstract index theorems. A survey of such index theorems is given and the concepts and ideas involved in these theorems are explained.
Author: Jean-Louis Loday Publisher: Springer Science & Business Media ISBN: 3662217392 Category : Mathematics Languages : en Pages : 467
Book Description
This book is a comprehensive study of cyclic homology theory together with its relationship with Hochschild homology, de Rham cohomology, S1 equivariant homology, the Chern character, Lie algebra homology, algebraic K-theory and non-commutative differential geometry. Though conceived as a basic reference on the subject, many parts of this book are accessible to graduate students.
Author: Joachim Cuntz Publisher: Springer Science & Business Media ISBN: 3764383992 Category : Mathematics Languages : en Pages : 268
Book Description
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: Joachim J. R. Cuntz Publisher: American Mathematical Soc. ISBN: 9780821871249 Category : Mathematics Languages : en Pages : 202
Book Description
Noncommutative geometry is a new field that is among the great challenges of present-day mathematics. Its methods allow one to treat noncommutative algebras - such as algebras of pseudodifferential operators, group algebras, or algebras arising from quantum field theory - on the same footing as commutative algebras, that is, as spaces. Applications range over many fields of mathematics and mathematical physics. This volume contains the proceedings of the workshop on "Cyclic Cohomology and Noncommutative Geometry" held at The Fields Institute (Waterloo, ON) in June 1995. The workshop was part of the program for the special year on operator algebras and its applications.
Author: A. Connes Publisher: American Mathematical Society ISBN: 1470469774 Category : Mathematics Languages : en Pages : 592
Book Description
This volume contains the proceedings of the virtual conference on Cyclic Cohomology at 40: Achievements and Future Prospects, held from September 27–October 1, 2021 and hosted by the Fields Institute for Research in Mathematical Sciences, Toronto, ON, Canada. Cyclic cohomology, since its discovery forty years ago in noncommutative differential geometry, has become a fundamental mathematical tool with applications in domains as diverse as analysis, algebraic K-theory, algebraic geometry, arithmetic geometry, solid state physics and quantum field theory. The reader will find survey articles providing a user-friendly introduction to applications of cyclic cohomology in such areas as higher categorical algebra, Hopf algebra symmetries, de Rham-Witt complex, quantum physics, etc., in which cyclic homology plays the role of a unifying theme. The researcher will find frontier research articles in which the cyclic theory provides a computational tool of great relevance. In particular, in analysis cyclic cohomology index formulas capture the higher invariants of manifolds, where the group symmetries are extended to Hopf algebra actions, and where Lie algebra cohomology is greatly extended to the cyclic cohomology of Hopf algebras which becomes the natural receptacle for characteristic classes. In algebraic topology the cyclotomic structure obtained using the cyclic subgroups of the circle action on topological Hochschild homology gives rise to remarkably significant arithmetic structures intimately related to crystalline cohomology through the de Rham-Witt complex, Fontaine's theory and the Fargues-Fontaine curve.
Author: Christian Groenbaek
Author: Christian Groenbaek
Author: Christian Grønbæk
Author: Christian Groenbaek
Author: Joachim Cuntz
Author: Ralf Meyer
Author: Joachim Cuntz
Author: Jean-Louis Loday
Author: Joachim Cuntz
Author: Joachim J. R. Cuntz
Author: A. Connes