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Cyclic Cohomology at 40: Achievements and Future Prospects
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 ca...
Operator Algebras
This book offers a comprehensive introduction to the general theory of C*-algebras and von Neumann algebras. Beginning with the basics, the theory is developed through such topics as tensor products, nuclearity and exactness, crossed products, K-theory, and quasidiagonality. The presentation carefully and precisely explains the main features of each part of the theory of operator algebras; most important arguments are at least outlined and many are presented in full detail.
Quanta of Maths
The work of Alain Connes has cut a wide swath across several areas of mathematics and physics. Reflecting its broad spectrum and profound impact on the contemporary mathematical landscape, this collection of articles covers a wealth of topics at the forefront of research in operator algebras, analysis, noncommutative geometry, topology, number theory and physics. Specific themes covered by the articles are as follows: entropy in operator algebras, regular $C^*$-algebras of integral domains, properly infinite $C^*$-algebras, representations of free groups and 1-cohomology, Leibniz seminorms and quantum metric spaces; von Neumann algebras, fundamental Group of $\mathrm{II}_1$ factors, subfacto...
Characterizations of C* Algebras
The first unified, in-depth discussion of the now classical Gelfand-Naimark theorems, thiscomprehensive text assesses the current status of modern analysis regarding both Banachand C*-algebras.Characterizations of C*-Algebras: The Gelfand-Naimark Theorems focuses on general theoryand basic properties in accordance with readers' needs ... provides complete proofs of theGelfand-Naimark theorems as well as refinements and extensions of the original axioms. . . gives applications of the theorems to topology, harmonic analysis. operator theory.group representations, and other topics ... treats Hermitian and symmetric *-algebras.algebras with and without identity, and algebras with arbitrary (poss...
K-theory and Noncommutative Geometry
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 quest...
Topics in Noncommutative Geometry
Luis Santalo Winter Schools are organized yearly by the Mathematics Department and the Santalo Mathematical Research Institute of the School of Exact and Natural Sciences of the University of Buenos Aires (FCEN). This volume contains the proceedings of the third Luis Santalo Winter School which was devoted to noncommutative geometry and held at FCEN July 26-August 6, 2010. Topics in this volume concern noncommutative geometry in a broad sense, encompassing various mathematical and physical theories that incorporate geometric ideas to the study of noncommutative phenomena. It explores connections with several areas including algebra, analysis, geometry, topology and mathematical physics. Burs...
Operator Algebras and Their Applications II
The study of operator algebras, which grew out of von Neumann's work in the 1920s and 30s on modelling quantum mechanics, has in recent years experienced tremendous growth and vitality, with significant applications in other areas both within mathematics and in other fields. For this reason, and because of the existence of a strong Canadian school in the subject, the topic was a natural candidate for an emphasis year at The Fields Institute. This volume is the second selection of papers that arose from the seminars and workshops of a year-long program, Operator Algebras and Applications, that took place at The Fields Institute. Topics covered include the classification of amenable C*-algebras, lifting theorems for completely positive maps, and automorphisms of von Neumann algebras of type III.
Mathematical Physics X
th This volume contains the proceedings of the X Congress of the Interna tional Association of Mathematical Physics, held at the University of Leipzig from 30 July until 9 August 1991. There were more than 400 participants, from 29 countries, making it a truly international gathering. The congress had the support of the Deutsche Forschungsgemeinschaft, the European Economic Community, the International Association of Math ematical Physics, the International Mathematical Union and the Interna tional Union of Pure and Applied Physics. There were also sponsors from in dustry and commerce: ATC Mann, Deutsche Bank AG, Miele & Cie GmbH, NEC Deutschland GmbH, Rank Xerox, Siemens AG and Stiftungsfon...
Classification of Nuclear C*-Algebras. Entropy in Operator Algebras
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 facto...
