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 in both pure mathematics and mathematical physics. The theory 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 Vaughn Jones of subfactor theory, leading to remarkable connections with knot theory, 3-manifolds, quantum groups, and integrable systems in statistical mechanics and conformal field theory. This book, one of the first in the area, looks at these combinatorial-algebraic developments from the perspective of operator algebras. With minimal prerequisites from classical theory, it brings the reader to the forefront of research.

Quantum Symmetries on Operator Algebras (Oxford Mathematical Monographs)

Science & Math

Kawahigashi, Yasuyuki

Evans, David E.

Mathematics

Oxford University Press

From Vertex Operator Algebras to Conformal Nets and Back (Memoirs of the American Mathematical Society)

Longo, Roberto

Carpi, Sebastiano

Weiner, Mihaly

Amer Mathematical Society

C* tensor categories are a point of contact where Operator Algebras and Quantum Field Theory meet. They are the underlying unifying concept for homomorphisms of (properly infinite) von Neumann algebras and representations of quantum observables.
The present introductory text reviews the basic notions and their cross-relations in different contexts. The focus is on Q-systems that serve as complete invariants, both for subfactors and for extensions of quantum field theory models.
It proceeds with various operations on Q-systems (several decompositions, the mirror Q-system, braided product, centre and full centre of Q-systems) some of which are defined only in the presence of a braiding.The last chapter gives a brief exposition of the relevance of the mathematical structures presented in the main body for applications in Quantum Field Theory (in particular two-dimensional Conformal Field Theory, also with boundaries or defects).

Tensor Categories and Endomorphisms of von Neumann Algebras: with Applications to Quantum Field Theory (SpringerBriefs in Mathematical Physics, 3)

Bischoff, Marcel

Rehren, Karl-Henning

Springer

This workshop gives an overview of the foremost developments in subfactor theory and related topics.

Subfactors: Proceedings of the Taniguchi Symposium on Operator Algebras

Araki, Huzihiro

World Scientific Pub Co Inc

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