Quantum Symmetries on Operator Algebras
Published on 21. May 1998
Book
Hardback
846 pages
978-0-19-851175-5 (ISBN)
Description
In the last 20 years, the study of operator algebras has developed from a branch of functional analysis to a central field of mathematics with applications and connections with different areas in both pure mathematics (foliations, index theory, K-theory, cyclic homology, affine Kac--Moody algebras, quantum groups, low dimensional topology) and mathematical physics (integrable theories, statistical mechanics, conformal field theories and the string theories of elementary particles). The theory of operator algebras was initiated by von Neumann and Murray as a tool for studying group representations and as a framework for quantum mechanics, and has since kept in touch with its roots in physics as a framework for quantum statistical mechanics and the formalism of algebraic quantum field theory. However, in 1981, the study of operator algebras took a new turn with the introduction by Vaughan Jones of subfactor theory and remarkable connections were found with knot theory, 3-manifolds, quantum groups and integrable systems in statistical mechanics and conformal field theory. The purpose of this book, one of the first in the area, is to look at these combinatorial-algebraic developments from the perspective of operator algebras; to bring the reader to the frontline of research with the minimum of prerequisites from classical theory.
Reviews / Votes
'This is a most remarkable book, and it certainly gives a hand to many mathematician, researchers as well as graduate students. On less than 850 pages, the author present vital information about 124 subjects in 15 chapters, in a manner accessible to any broadly trained analyst.....As a collection of broad and relevant information the book bears some similarity to the famous volumes of Dunford and Scwarz' Zentralblatt Mathematik 'This monumental book presents some of the most exciting developments of the past 15 years within operator algebra theory and its interaction with mathematical physics and algebraic topology...In the preface the authors state:this is aimed at graduate students and researchers in operator algebras, and statistical mechanics, algebraic, topolgical and conformal field theory and low dimensional topological invarieants. All ot these will find a rich source of study and references in this extensively annotated and documented volume, and not least a wide range of new perspectives on their own subject.' Mathematical Reviews ' This monumental book presents some of the most exciting developments of the past 15 years within operator algebra theory and its interaction with mathematical physics and algebraic topology ... a rich source of study and references in this extensively annotated and documented volume, and not least a wide range of new perspectives on their own subject' MathsSciNetMore details
Series
Language
English
Place of publication
Oxford
United Kingdom
Target group
Professional and scholarly
Illustrations
numerous line figures
Dimensions
Height: 241 mm
Width: 161 mm
Thickness: 48 mm
Weight
1337 gr
ISBN-13
978-0-19-851175-5 (9780198511755)
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Schweitzer Classification
Persons
(until early 1998) Professor David E. Evans, Professor of Mathematics, Department of Mathematics, University of Wales, Swansea, Singleton Park, Swansea, SA2 8PP, WALES. Tel: +44 1792 295460; fax: +44 1792 295843; email: David.E.Evans@swansea.ac.uk (from early 1998) Mathematics Institute, University of Wales, Cardiff, Senghennydd Road, Cardiff CF2 4YH Tel: +44 1222 874827; fax: +44 1222 874199 Professor Yasuyuki Kawahigashi, Associate Professor, Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo, 153, JAPAN. Tel: +81 3 5465 7024; fax: +81 3 5465 7012; email: yasuyuki@ms.u-tokyo.ac.jp
Content
1. Operator theory basics ; 2. C*-algebra basics ; 3. K-theory ; 4. Positivity and semigroups ; 5. von Neumann algebra basics ; 6. The Fermion algebra ; 7. The Ising model ; 8. Conformal field theory ; 9. Subfactors and bimodules ; 10. Axiomatization of paragroups ; 11. String algebras and flat connections ; 12. Topological quantum field theory (TQFT) and paragroups ; 13. Rational conformal field theory (RCFT) and paragroups ; 14. Commuting squares of II1 factors ; 15. Automorphisms of subfactors and central sequences ; Bibliography ; Index