Book•
Theory of Operator Algebras II
01 Jan 1979-
About: The article was published on 1979-01-01 and is currently open access. It has received 3776 citations till now. The article focuses on the topics: Nest algebra & Ladder operator.
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1,793 citations
TL;DR: In this paper, a construction that yields a class of translation invariant states on quantum spin chains, characterized by the property that the correlations across any bond can be modeled on a finite-dimensional vector space.
Abstract: We study a construction that yields a class of translation invariant states on quantum spin chains, characterized by the property that the correlations across any bond can be modeled on a finite-dimensional vector space. These states can be considered as generalized valence bond states, and they are dense in the set of all translation invariant states. We develop a complete theory of the ergodic decomposition of such states, including the decomposition into periodic “Neel ordered” states. The ergodic components have exponential decay of correlations. All states considered can be obtained as “local functions” of states of a special kind, so-called “purely generated states,” which are shown to be ground states for suitably chosen finite range VBS interactions. We show that all these generalized VBS models have a spectral gap. Our theory does not require symmetry of the state with respect to a local gauge group. In particular we illustrate our results with a one-parameter family of examples which are not isotropic except for one special case. This isotropic model coincides with the one-dimensional antiferromagnet, recently studied by Affleck, Kennedy, Lieb, and Tasaki.
1,308 citations
Cites background from "Theory of Operator Algebras II"
...symbol "| will always refer to the minimal C*-tensor product [ 62 ]....
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...It is well-known that a map of the form X~-~ V*XVi is completely positive [ 62 ]....
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Book•
01 Jan 2008TL;DR: In this article, the authors present a classification of group von Neumann algebras based on the following properties: weak expectation property, local lifting property, and local reflexivity.
Abstract: Fundamental facts Basic theory: Nuclear and exact $\textrm{C}^*$-algebras: Definitions, basic facts and examples Tensor products Constructions Exact groups and related topics Amenable traces and Kirchberg's factorization property Quasidiagonal C*-algebras AF embeddablity Local reflexivity and other tensor product conditions Summary and open problems Special topics: Simple $\textrm{C}^*$-algebras Approximation properties for groups Weak expectation property and local lifting property Weakly exact von Neumann algebras Applications: Classification of group von Neumann algebras Herrero's approximation problem Counterexamples in $\textrm{K}$-homology and $\textrm{K}$-theory Appendices: Ultrafilters and ultraproducts Operator spaces, completely bounded maps and duality Lifting theorems Positive definite functions, cocycles and Schoenberg's Theorem Groups and graphs Bimodules over von Neumann algebras Bibliography Notation index Subject index.
1,079 citations
760 citations
TL;DR: The theory of locally compact quantum groups that are studied in the framework of operator algebras, i.e., C*-alges and von Neumann alges, is introduced in this paper.
Abstract: These lecture notes are intended as an introduction to the theory of locally compact quantum groups that are studied in the framework of operator algebras, i.e. C*-algebras and von Neumann algebras. The presentation revolves around the definition of a locally compact quantum group as given in [KuV00a] and [KuV03].
609 citations