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Operator algebra
About: Operator algebra is a research topic. Over the lifetime, 5783 publications have been published within this topic receiving 165303 citations.
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TL;DR: In this paper, three classes of nonlinear integrable equations in 1 + 1 and 2 + 1 dimensions are discussed within the framework of generalized Lax equations and Sato's approach.
Abstract: Three different decompositions of the algebra of pseudo-differential operators and the corresponding r-matrices are considered. Three associated classes of nonlinear integrable equations in 1 +1 and 2 + 1 dimensions are discussed within the framework of generalized Lax equations and Sato's approach. The 2 +1-dimensional hierarchies are associated with the Kadomtsev-Petviashvili (KP) equation, the modified KP equation and a Dym equation, respectively. Reductions of the general hierarchies lead to other known integrable 2 + 1dimensional equations as well as to a variety of integrable equations in 1 +1 dimensions. It is shown, how the multi-Hamiltonian structure of the 1 + 1-dimensional equations can be obtained from the underlying r-matrices. Further, intimate relations between the equations associated with the three different r-matrices are revealed. The three classes are related by Darboux theorems originating from gauge transformations and reciprocal links of the Lax operators. These connections are discussed on a general level, leading to a unified picture on (reciprocal) Backlund and auto-Backlund transformations for large classes of integrable equations covered by the KP, the modified KP, and the Dym hierarchies. §
116 citations
01 Mar 1996
TL;DR: In this article, a 4D analog of 2D WZW theory is investigated, and the theory turns out to have surprising finiteness properties and an infinite-dimensional current algebra symmetry.
Abstract: We investigate a 4D analog of 2D WZW theory. The theory turns out to have surprising finiteness properties and an infinite-dimensional current algebra symmetry. Some correlation functions are determined by this symmetry. One way to define the theory systematically proceeds by the quantization of moduli spaces of holomorphic vector bundles over algebraic surfaces. We outline how one can define vertex operators in the theories. Finally, we define four-dimensional “conformal blocks” and present an analog of the Verlinde formula.
115 citations
TL;DR: In this article, it was shown that every von Neumann algebra is completely rational iff Ω(I) and ℬ(I)-is completely rational under a coset model.
Abstract: Given an irreducible local conformal net 𝒜 of von Neumann algebras on S
1
and a finite-index conformal subnet ℬ⊂𝒜, we show that 𝒜 is completely rational iff ℬ is completely rational. In particular this extends a result of F. Xu for the orbifold construction. By applying previous results of Xu, many coset models turn out to be completely rational and the structure results in [27] hold. Our proofs are based on an analysis of the net inclusion ℬ⊂𝒜; among other things we show that, for a fixed interval I, every von Neumann algebra intermediate between ℬ(I) and 𝒜(I) comes from an intermediate conformal net ℒ between ℬ and 𝒜 with ℒ(I)=. We make use of a theorem of Watatani (type II case) and Teruya and Watatani (type III case) on the finiteness of the set ℑ(𝒩,ℳ) of intermediate subfactors in an irreducible inclusion of factors 𝒩⊂ℳ with finite Jones index [ℳ:𝒩]. We provide a unified proof of this result that gives in particular an explicit bound for the cardinality of ℑ(𝒩,ℳ) which depends only on [ℳ:𝒩].
115 citations
TL;DR: In this article, a characterization of additive mappings on the subalgebra of all finite-rank operators which preserve operators of rank one or projections of rank two is given. In the real case such mappings are automatically linear.
114 citations
01 Jan 2011
TL;DR: In this paper, a prehistory of n-categorical physics is described, and the strong free-will theorem is proved in algebraic quantum field theory, where locality and operational independence are discussed.
Abstract: Part I. Beyond the Hilbert Space Formalism: Category Theory: 1. A prehistory of n-categorical physics John C. Baez and Aaron Lauda 2. A universe of processes and some of its guises Bob Coecke 3. Topos methods in the foundations of physics Chris J. Isham 4. The physical interpretation of daseinisation Andreas Doring 5. Classical and quantum observables Hans F. de Groote 6. Bohrification Chris Heunen, Nicolaas P. Landsman and Bas Spitters Part II. Beyond the Hilbert Space Formalism: Operator Algebras: 7. Yet more ado about nothing: the remarkable relativistic vacuum state Stephen J. Summers 8. Einstein meets von Neumann: locality and operational independence in algebraic quantum field theory Miklos Redei Part III. Behind the Hilbert Space Formalism: 9. Quantum theory and beyond: is entanglement special? Borivoje Dakic and Caslav Brukner 10. Is Von Neumann's 'no hidden variables' proof silly? Jeffrey Bub 11. Foliable operational structures for general probabilistic theories Lucien Hardy 12. The strong free will theorem John H. Conway and Simon Kochen.
113 citations