Operator Algebras And Geometry

In the early 1980's topologists and geometers for the first time came across unfamiliar words like $C^*$-algebras and von Neumann algebras through the discovery of new knot invariants (by V. F. R. Jones) or through a remarkable result on the relationship between characteristic classes of foliations and the types of certain von Neumann algebras. During the following two decades, a great deal of progress was achieved in studying the interaction between geometry and analysis, in particular in noncommutative geometry and mathematical physics. The present book provides an overview of operator algebra theory and an introduction to basic tools used in noncommutative geometry. The book concludes with applications of operator algebras to Atiyah-Singer type index theorems. The purpose of the book is to convey an outline and general idea of operator algebra theory, to some extent focusing on examples. The book is aimed at researchers and graduate students working in differential topology, differential geometry, and global analysis who are interested in learning about operator algebras.

In this book we give a complete geometric description of state spaces of operator algebras, Jordan as well as associative. That is, we give axiomatic characterizations of those convex sets that are state spaces of C*-algebras and von Neumann algebras, together with such characterizations for the normed Jordan algebras called JB-algebras and JBW-algebras. These non associative algebras generalize C*-algebras and von Neumann algebras re spectively, and the characterization of their state spaces is not only of interest in itself, but is also an important intermediate step towards the characterization of the state spaces of the associative algebras. This book gives a complete and updated presentation of the character ization theorems of [10]' [11] and [71]. Our previous book State spaces of operator algebras: basic theory, orientations and C*-products, referenced as [AS] in the sequel, gives an account of the necessary prerequisites on C*-algebras and von Neumann algebras, as well as a discussion of the key notion of orientations of state spaces. For the convenience of the reader, we have summarized these prerequisites in an appendix which contains all relevant definitions and results (listed as (AI), (A2), ... ), with reference back to [AS] for proofs, so that this book is self-contained.

These volumes form an authoritative statement of the current state of research in Operator Algebras. They consist of papers arising from a year-long symposium held at the University of Warwick. Contributors include many very well-known figures in the field.

This book is the proceeding of the International Workshop on Operator Theory and Applications (IWOTA) held in July 2018 in Shanghai, China. It consists of original papers, surveys and expository articles in the broad areas of operator theory, operator algebras and noncommutative topology. Its goal is to give graduate students and researchers a relatively comprehensive overview of the current status of research in the relevant fields. The book is also a special volume dedicated to the memory of Ronald G. Douglas who passed away on February 27, 2018 at the age of 79. Many of the contributors are Douglas’ students and past collaborators. Their articles attest and commemorate his life-long contribution and influence to these fields.

The theory of vertex operator algebras and their representations has been showing its power in the solution of concrete mathematical problems and in the understanding of conceptual but subtle mathematical and physical struc- tures of conformal field theories. Much of the recent progress has deep connec- tions with complex analysis and conformal geometry. Future developments, especially constructions and studies of higher-genus theories, will need a solid geometric theory of vertex operator algebras. Back in 1986, Manin already observed in Man) that the quantum theory of (super )strings existed (in some sense) in two entirely different mathematical fields. Under canonical quantization this theory appeared to a mathematician as the representation theories of the Heisenberg, Vir as oro and affine Kac- Moody algebras and their superextensions. Quantization with the help of the Polyakov path integral led on the other hand to the analytic theory of algebraic (super ) curves and their moduli spaces, to invariants of the type of the analytic curvature, and so on.He pointed out further that establishing direct mathematical connections between these two forms of a single theory was a big and important problem. On the one hand, the theory of vertex operator algebras and their repre- sentations unifies (and considerably extends) the representation theories of the Heisenberg, Virasoro and Kac-Moody algebras and their superextensions.

John von Neumann and Marshall Stone were two giants of Twentieth Century mathematics. In honor of the 100th anniversary of their births, a mathematical celebration was organized featuring developments in fields where both men were major influences. This volume contains articles from the AMS Special Session, Operator Algebras, Quantization and Noncommutative Geometry: A Centennial Celebration in Honor of John von Neumann and Marshall H. Stone.Papers range from expository and historical surveys to original research articles. All articles were carefully refereed and cover a broad range of mathematical topics reflecting the fundamental ideas of von Neumann and Stone. Most contributions are expanded versions of the talks and were written exclusively for this volume. Included, among others, are articles by George W. Mackey, Nigel Higson, and Marc Rieffel. Also featured is a reprint of P.R. Halmos' ""The Legend of John von Neumann"". The book is suitable for graduate students and researchers interested in operator algebras and applications, including noncommutative geometry.

Mathematics for infinite dimensional objects is becoming more and more important today both in theory and application. Rings of operators, renamed von Neumann algebras by J. Dixmier, were first introduced by J. von Neumann fifty years ago, 1929, in [254] with his grand aim of giving a sound founda tion to mathematical sciences of infinite nature. J. von Neumann and his collaborator F. J. Murray laid down the foundation for this new field of mathematics, operator algebras, in a series of papers, [240], [241], [242], [257] and [259], during the period of the 1930s and early in the 1940s. In the introduction to this series of investigations, they stated Their solution 1 {to the problems of understanding rings of operators) seems to be essential for the further advance of abstract operator theory in Hilbert space under several aspects. First, the formal calculus with operator-rings leads to them. Second, our attempts to generalize the theory of unitary group-representations essentially beyond their classical frame have always been blocked by the unsolved questions connected with these problems. Third, various aspects of the quantum mechanical formalism suggest strongly the elucidation of this subject. Fourth, the knowledge obtained in these investigations gives an approach to a class of abstract algebras without a finite basis, which seems to differ essentially from all types hitherto investigated. Since then there has appeared a large volume of literature, and a great deal of progress has been achieved by many mathematicians.

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.