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Characterizing Groupoid C*-algebras of Non-Hausdorff Étale Groupoids

2022-10-18 Mathematics
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Release : 2022-10-18
Genre : Mathematics
Kind : eBook
Book Rating : 136/5 ( reviews)

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Book Synopsis Characterizing Groupoid C*-algebras of Non-Hausdorff Étale Groupoids by : Ruy Exel

Download or read book Characterizing Groupoid C*-algebras of Non-Hausdorff Étale Groupoids written by Ruy Exel. This book was released on 2022-10-18. Available in PDF, EPUB and Kindle. Book excerpt: This book develops tools to handle C*-algebras arising as completions of convolution algebras of sections of line bundles over possibly non-Hausdorff groupoids. A fundamental result of Gelfand describes commutative C*-algebras as continuous functions on locally compact Hausdorff spaces. Kumjian, and later Renault, showed that Gelfand's result can be extended to include non-commutative C*-algebras containing a commutative C*-algebra. In their setting, the C*-algebras in question may be described as the completion of convolution algebras of functions on twisted Hausdorff groupoids with respect to a certain norm. However, there are many natural settings in which the Kumjian–Renault theory does not apply, in part because the groupoids which arise are not Hausdorff. In fact, non-Hausdorff groupoids have been a source of surprising counterexamples and technical difficulties for decades. Including numerous illustrative examples, this book extends the Kumjian–Renault theory to a much broader class of C*-algebras. This work will be of interest to researchers and graduate students in the area of groupoid C*-algebras, the interface between dynamical systems and C*-algebras, and related fields.

Tool Kit for Groupoid C∗ -Algebras

2019-09-24 C*-algebras
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Release : 2019-09-24
Genre : C*-algebras
Kind : eBook
Book Rating : 336/5 ( reviews)

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Book Synopsis Tool Kit for Groupoid C∗ -Algebras by : Dana P. Williams

Download or read book Tool Kit for Groupoid C∗ -Algebras written by Dana P. Williams. This book was released on 2019-09-24. Available in PDF, EPUB and Kindle. Book excerpt: The construction of a C∗-algebra from a locally compact groupoid is an important generalization of the group C∗-algebra construction and of the transformation group C∗-algebra construction. Since their introduction in 1980, groupoid C∗-algebras have been intensively studied with diverse applications, including graph algebras, classification theory, variations on the Baum-Connes conjecture, and noncommutative geometry. This book provides a detailed introduction to this vast subject and is suitable for graduate students or any researcher who wants to use groupoid C∗-algebras in their work. The main focus is to equip the reader with modern versions of the basic technical tools used in the subject, which will allow the reader to understand fundamental results and make contributions to various areas in the subject. Thus, in addition to covering the basic properties and construction of groupoid C∗-algebras, the focus is to give a modern treatment of some of the major developments in the subject in recent years, including the Equivalence Theorem and the Disintegration Theorem. Also covered are the complicated subjects of amenability of groupoids and simplicity results. The book is reasonably self-contained and accessible to graduate students with a good background in operator algebras.

Operator Algebras and Dynamics: Groupoids, Crossed Products, and Rokhlin Dimension

2020-06-22 Mathematics
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Release : 2020-06-22
Genre : Mathematics
Kind : eBook
Book Rating : 130/5 ( reviews)

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Book Synopsis Operator Algebras and Dynamics: Groupoids, Crossed Products, and Rokhlin Dimension by : Aidan Sims

Download or read book Operator Algebras and Dynamics: Groupoids, Crossed Products, and Rokhlin Dimension written by Aidan Sims. This book was released on 2020-06-22. Available in PDF, EPUB and Kindle. Book excerpt: 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.

A Groupoid Approach to C*-Algebras

2006-11-15 Mathematics
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Release : 2006-11-15
Genre : Mathematics
Kind : eBook
Book Rating : 181/5 ( reviews)

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Book Synopsis A Groupoid Approach to C*-Algebras by : Jean Renault

Download or read book A Groupoid Approach to C*-Algebras written by Jean Renault. This book was released on 2006-11-15. Available in PDF, EPUB and Kindle. Book excerpt:

Cyclic Cohomology at 40: Achievements and Future Prospects

2023-02-23 Mathematics
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Release : 2023-02-23
Genre : Mathematics
Kind : eBook
Book Rating : 774/5 ( reviews)

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Book Synopsis Cyclic Cohomology at 40: Achievements and Future Prospects by : A. Connes

Download or read book Cyclic Cohomology at 40: Achievements and Future Prospects written by A. Connes. This book was released on 2023-02-23. Available in PDF, EPUB and Kindle. Book excerpt: 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.

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