Showing papers on "Operator algebra published in 2005"
TL;DR: In this article, the quantum deformations of cluster algebras are studied in the context of total positivity and canonical bases in semisimple groups and their quantum analogs.
363 citations
TL;DR: A systematic framework for noncommutative (NC) quantum field theory (QFT) within the new concept of relativistic invariance based on the notion of twisted Poincare symmetry is presented and the NC analog of Haag's theorem is proved.
Abstract: We present a systematic framework for noncommutative (NC) quantum field theory (QFT) within the new concept of relativistic invariance based on the notion of twisted Poincare symmetry, as proposed by Chaichian et al. [Phys. Lett. B 604, 98 (2004)]. This allows us to formulate and investigate all fundamental issues of relativistic QFT and offers a firm frame for the classification of particles according to the representation theory of the twisted Poincare symmetry and as a result for the NC versions of CPT and spin-statistics theorems, among others, discussed earlier in the literature. As a further application of this new concept of relativism we prove the NC analog of Haag's theorem.
286 citations
TL;DR: A proof of the Verlinde conjecture for V is announced of the statement that the matrices formed by the fusion rules among irreducible V-modules are diagonalized by the matrix given by the action of the modular transformation tau |--> -1/tau on the space of characters of irreducing V- modules.
Abstract: Let V be a simple vertex operator algebra satisfying the following conditions: (i) V ( n ) = 0 for n < 0, , and the contragredient module V' is isomorphic to V as a V-module; (ii) every weak V-module is completely reducible; (iii) V is C 2-cofinite. We announce a proof of the Verlinde conjecture for V, that is, of the statement that the matrices formed by the fusion rules among irreducible V-modules are diagonalized by the matrix given by the action of the modular transformation τ → –1/τ on the space of characters of irreducible V-modules. We discuss some consequences of the Verlinde conjecture, including the Verlinde formula for the fusion rules, a formula for the matrix given by the action of τ → –1/τ, and the symmetry of this matrix. We also announce a proof of the rigidity and nondegeneracy property of the braided tensor category structure on the category of V-modules when V satisfies in addition the condition that irreducible V-modules not equivalent to V have no nonzero elements of weight 0. In particular, the category of V-modules has a natural structure of modular tensor category.
171 citations
TL;DR: In this article, the complete spectral curve for an arbitrary local operator, including fermions and covariant derivatives, of one-loop N=4 gauge theory in the thermodynamic limit was constructed.
Abstract: We construct the complete spectral curve for an arbitrary local operator, including fermions and covariant derivatives, of one-loop N=4 gauge theory in the thermodynamic limit. This curve perfectly reproduces the Frolov-Tseytlin limit of the full spectral curve of classical strings on AdS_5xS^5 derived in hep-th/0502226. To complete the comparison we introduce stacks, novel bound states of roots of different flavors which arise in the thermodynamic limit of the corresponding Bethe ansatz equations. We furthermore show the equivalence of various types of Bethe equations for the underlying su(2,2|4) superalgebra, in particular of the type "Beauty" and "Beast".
171 citations
TL;DR: In this paper, it was shown that if every module W for a vertex operator algebra V = ∐n∈ℤV(n) satisfies the condition dim W/C1(W) 0 V(n), and w ∈ W, then matrix elements of products and iterates of intertwining operators satisfy certain systems of differential equations.
Abstract: We show that if every module W for a vertex operator algebra V = ∐n∈ℤV(n) satisfies the condition dim W/C1(W) 0 V(n) and w ∈ W, then matrix elements of products and iterates of intertwining operators satisfy certain systems of differential equations. Moreover, for prescribed singular points, there exist such systems of differential equations such that the prescribed singular points are regular. The finiteness of the fusion rules is an immediate consequence of a result used to establish the existence of such systems. Using these systems of differential equations and some additional reductivity conditions, we prove that products of intertwining operators for V satisfy the convergence and extension property needed in the tensor product theory for V-modules. Consequently, when a vertex operator algebra V satisfies all the conditions mentioned above, we obtain a natural structure of vertex tensor category (consequently braided tensor category) on the category of V-modules and a natural structure of intertwining operator algebra on the direct sum of all (inequivalent) irreducible V-modules.
165 citations
Journal Article•
TL;DR: In this paper, it was shown that a family of representations (in Agler's sense) of either an opera-tor algebra or an operator space has boundary representations, which leads to a direct proof of Hamana's result that all unital operator algebras have enough such boundary representations to generate the C ⁄ -envelope.
Abstract: In analogy with the peak points of the Shilov boundary of a uni- form algebra, Arveson defined the notion of boundary representations among the completely contractive representations of a unital operator algebra. How- ever, he was unable to show that such representations always exist. Drop- ping his original condition that such representations should be irreducible, we show that a family of representations (in Agler's sense) of either an opera- tor algebra or an operator space has boundary representations. This leads to a direct proof of Hamana's result that all unital operator algebras have enough such boundary representations to generate the C ⁄ -envelope.
148 citations
TL;DR: In this article, the authors studied the geometry of the space of Hermitian operators on a Hilbert space and showed that rank-k states, k = 1,..., n, are a smooth manifold of real dimension 2nk? k2? 1 and this stratification is maximal in the sense that every smooth curve in, viewed as a subset of the dual to the Lie algebra of the unitary group, at every point must be tangent to the strata it crosses.
Abstract: Various problems concerning the geometry of the space of Hermitian operators on a Hilbert space are addressed. In particular, we study the canonical Poisson and Riemann?Jordan tensors and the corresponding foliations into K?hler submanifolds. It is also shown that the space of density states on an n-dimensional Hilbert space is naturally a manifold stratified space with the stratification induced by the the rank of the state. Thus the space of rank-k states, k = 1, ..., n, is a smooth manifold of (real) dimension 2nk ? k2 ? 1 and this stratification is maximal in the sense that every smooth curve in , viewed as a subset of the dual to the Lie algebra of the unitary group , at every point must be tangent to the strata it crosses. For a quantum composite system, i.e. for a Hilbert space decomposition , an abstract criterion of entanglement is proved.
148 citations
TL;DR: In this paper, a new family of quantum integrable models is proposed, which is generated by a dual pair of operators { A, A ∗ ∈ A subject to q-deformed Dolan-Grady relations.
124 citations
TL;DR: In this article, a one-loop dilatation operator for light-cone was shown to have universal integrability properties in all (super)symmetric Yang-Mills theories on the lightcone ranging from gluodynamics all the way to the maximally supersymmetric N = 4 theory.
98 citations
TL;DR: In this paper, a simple calculus for calculations in the universal enveloping algebra is presented, where the algebra of differential geometry operations on symmetric tensors over constant curvature manifolds forms a novel deformation of the sl ( 2, R ) ⋉ R 2 Lie algebra.
96 citations
TL;DR: The fusion rules for vertex operator algebras M(1)+ (of any rank) and V+ (for any positive definite even lattice L) are determined completely as discussed by the authors.
Abstract: The fusion rules for the vertex operator algebras M(1)+ (of any rank) and V+
L
(for any positive definite even lattice L) are determined completely.
TL;DR: For quantum systems with a finite number of degrees of freedom, the simplest possibility, i.e., factors of type I in the terminology of Murray and von Neumann, are perfectly adequate.
TL;DR: In this article, it was shown that the large Nc dynamics of the parent theory, restricted to the subspace invariant under the orbifold projection symmetry, and the daughter theory, restricting to the untwisted sector invariant and theory space permutation symmetry, coincide.
Abstract: Large N coherent state methods are used to study the relation between U(Nc) gauge theories containing adjoint representation matter fields and their orbifold projections. The classical dynamical systems which reproduce the large Nc limits of the quantum dynamics in parent and daughter orbifold theories are compared. We demonstrate that the large Nc dynamics of the parent theory, restricted to the subspace invariant under the orbifold projection symmetry, and the large Nc dynamics of the daughter theory, restricted to the untwisted sector invariant under ``theory space'' permutations, coincide. This implies equality, in the large Nc limit, between appropriately identified connected correlation functions in parent and daughter theories, provided the orbifold projection symmetry is not spontaneously broken in the parent theory and the theory space permutation symmetry is not spontaneously broken in the daughter. The necessity of these symmetry realization conditions for the validity of the large Nc equivalence is unsurprising, but demonstrating the sufficiency of these conditions is new. This work extends an earlier proof of non-perturbative large Nc equivalence which was only valid in the phase of the (lattice regularized) theories continuously connected to large mass and strong coupling [1].
TL;DR: In this article, the representations of the algebra of coordinates and momenta of non-commutative phase space are given, and the map of Schrodinger equation from non-complementary space to commutative space is obtained.
Abstract: The representations of the algebra of coordinates and momenta of noncommutative phase space are given. We study, as an example, the harmonic oscillator in noncommutative space of any dimension. Finally the map of Schrodinger equation from noncommutative space to commutative space is obtained.
TL;DR: For the Dirac operator D on the standard quantum sphere, the authors obtained an asymptotic expansion of the SU q (2)-equivariant entire cyclic cocycle corresponding to when evaluated on the element.
Abstract: For the Dirac operator D on the standard quantum sphere we obtain an asymptotic expansion of the SU
q
(2)-equivariant entire cyclic cocycle corresponding to when evaluated on the element The constant term of this expansion is a twisted cyclic cocycle which up to a scalar coincides with the volume form and computes the quantum as well as the classical Fredholm indices.
TL;DR: For locally compact groups G and H, this article showed that if G is amenable then any completely bounded homomorphism Φ : A (G ) → B (H ) is of this form; and this theorem fails if G contains a discrete nonabelian free group.
TL;DR: A number of Application Program Interfaces for coding linear algebra algorithms for the MATLAB M-script and C programming languages appear to be simple, almost trivial, extensions of those languages, but with them, the task of programming and maintaining families of algorithms for a broad spectrum of linear algebra operations is greatly simplified.
Abstract: In this article, we present a number of Application Program Interfaces (APIs) for coding linear algebra algorithms. On the surface, these APIs for the MATLAB M-script and C programming languages appear to be simple, almost trivial, extensions of those languages. Yet with them, the task of programming and maintaining families of algorithms for a broad spectrum of linear algebra operations is greatly simplified. In combination with our Formal Linear Algebra Methods Environment (FLAME) approach to deriving such families of algorithms, dozens of algorithms for a single linear algebra operation can be derived, verified to be correct, implemented, and tested, often in a matter of minutes per algorithm. Since the algorithms are expressed in code much like they are explained in a classroom setting, these APIs become not just a tool for implementing libraries, but also a valuable tool for teaching the algorithms that are incorporated in the libraries. In combination with an extension of the Parallel Linear Algebra Package (PLAPACK) API, the approach presents a migratory path from algorithm to MATLAB implementation to high-performance sequential implementation to parallel implementation. Finally, the APIs are being used to create a repository of algorithms and implementations for linear algebra operations, the FLAME Interface REpository (FIRE), which already features hundreds of algorithms for dozens of commonly encountered linear algebra operations.
TL;DR: In this article, the existence and associativity of the non-meromorphic operator product expansion for an infinite family of vertex operator algebras, the triplet W-algebra, using results from P(z)-tensor product theory was proved.
Abstract: We prove the existence and associativity of the nonmeromorphic operator product expansion for an infinite family of vertex operator algebras, the triplet W-algebras, using results from P(z)-tensor product theory. While doing this, we also show that all these vertex operator algebras are C_2-cofinite.
TL;DR: In this paper, it was shown that the trace functions associated to the twisted sectors are holomorphic in the upper half plane for any commuting pairs in G under the C2-cofinite condition.
Abstract: This paper is about the orbifold theory for vertex operator superalgebras. Given a vertex operator superalgebra V and a finite automorphism group G of V, we show that the trace functions associated to the twisted sectors are holomorphic in the upper half plane for any commuting pairs in G under the C2-cofinite condition. We also establish that these functions afford a representation of the full modular group if V is C2-cofinite and g-rational for any g ∈ G.
Posted Content•
TL;DR: In this article, it was shown that the ideal of a general operator space X is the intersection of X with an ideal in any C *-algebra or C*-module.
Abstract: In recent work of the second author, a technical result was proved establishing a bijective correspondence between certain open projections in a C*-algebra containing an operator algebra A, and certain one-sided ideals of A. Here we give several remarkable consequences of this result. These include a generalization of the theory of hereditary subalgebras of a C*-algebra, and the solution of a ten year old problem on the Morita equivalence of operator algebras. In particular, the latter gives a very clean generalization of the notion of Hilbert C*-modules to nonselfadjoint algebras. We show that an `ideal' of a general operator space X is the intersection of X with an `ideal' in any containing C*-algebra or C*-module. Finally, we discuss the noncommutative variant of the classical theory of `peak sets'.
TL;DR: In this article, it was shown that Witten-Dijkgraaf-Verlinde equation underlies the construction of N = 4 superconformal multi-particle mechanics in one dimension.
Abstract: We show that Witten-Dijkgraaf-Verlinde-Verlinde equation underlies the construction of N=4 superconformal multi--particle mechanics in one dimension, including a N=4 superconformal Calogero model.
TL;DR: In this article, it was shown that the value of the central charge c is an invariant and hence the Virasoro nets for different values of c are not isomorphic as Mobius covariant nets.
Abstract: A Mobius covariant net of von Neumann algebras on S1 is diffeomorphism covariant if its Mobius symmetry extends to diffeomorphism symmetry. We prove that in case the net is either a Virasoro net or any at least 4-regular net such an extension is unique: the local algebras together with the Mobius symmetry (equivalently: the local algebras together with the vacuum vector) completely determine it. We draw the two following conclusions for such theories. (1) The value of the central charge c is an invariant and hence the Virasoro nets for different values of c are not isomorphic as Mobius covariant nets. (2) A vacuum preserving internal symmetry always commutes with the diffeomorphism symmetries. We further use our result to give a large class of new examples of nets (even strongly additive ones), which are not diffeomorphism covariant; i.e. which do not admit an extension of the symmetry to Diff+(S1).
TL;DR: In this article, the properties of the topological space of weighted composition operators on the space of bounded analytic functions on the open unit disk in the uniform operator topology were studied.
Abstract: We study properties of the topological space of weighted composition operators on the space of bounded analytic functions on the open unit disk in the uniform operator topology. Moreover, we characterize the compactness of differences of two weighted composition operators.
TL;DR: In this article, a simple nuclear C-algebra A having a weakly unperforated K∗-group which does not absorb Z provides a counterexample to Elliott's conjecture that the simple nuclear Calgebras will be classified by the Elliott invariant.
Abstract: Jiang and Su and (independently) Elliott discovered a simple, nuclear, infinite-dimensional C-algebra Z having the same Elliott invariant as the complex numbers. For a nuclear C-algebra A with weakly unperforated K∗-group the Elliott invariant of A ⊗ Z is isomorphic to that of A. Thus, any simple nuclear C-algebra A having a weakly unperforated K∗-group which does not absorb Z provides a counterexample to Elliott’s conjecture that the simple nuclear Calgebras will be classified by the Elliott invariant. In the sequel we exhibit a separable, infinite-dimensional, stably finite instance of such a non-Z-absorbing algebra A, and so provide a counterexample to the Elliott conjecture for the class of simple, nuclear, infinite-dimensional, stably finite, separable C-algebras.
TL;DR: In this paper, it was shown that the canonical extension of a monotone bounded lattice expansion can be embedded in the MacNeille completion of any sufficiently saturated elementary extension of the original structure.
Abstract: Let V be a variety of monotone bounded lattice expansions, that is, bounded lattices endowed with additional operations, each of which is order preserving or reversing in each coordinate. We prove that if V is closed under MacNeille completions, then it is also closed under canonical extensions. As a corollary we show that in the case of Boolean algebras with operators, any such variety V is generated by an elementary class of relational structures. Our main technical construction reveals that the canonical extension of a monotone bounded lattice expansion can be embedded in the MacNeille completion of any sufficiently saturated elementary extension of the original structure.
Book•
01 Nov 2005
TL;DR: In this article, the authors defined the notion of equivariant monotone operators as reproducing kernels of H*-ideals of H *-algebras.
Abstract: TOPOLOGICAL LIE ALGEBRAS Fundamentals Universal enveloping algebras The Baker-Campbell-Hausdor series Convergence of the Baker-Campbell-Hausdor series Notes LIE GROUPS AND THEIR LIE ALGEBRAS Definition of Lie groups The Lie algebra of a Lie group Logarithmic derivatives The exponential map Special features of Banach-Lie groups Notes ENLARGIBILITY Integrating Lie algebra homomorphisms Topological properties of certain Lie groups Enlargible Lie algebras Notes Smooth Homogeneous Spaces Basic facts on smooth homogeneous spaces Symplectic homogeneous spaces Some homogeneous spaces related to operator algebras Notes QUASIMULTIPLICATIVE MAPS Supports, convolution, and quasimultiplicativity Separate parts of supports Hermitian maps Notes COMPLEX STRUCTURES ON HOMOGENEOUS SPACES General results Pseudo-Kahler manifolds Flag manifolds in Banach algebras Notes EQUIVARIANT MONOTONE OPERATORS Definition of equivariant monotone operators H*-algebras and L*-algebras Equivariant monotone operators as reproducing kernels H*-ideals of H*-algebras Elementary properties of H*-ideals Notes L*-IDEALS AND EQUIVARIANT MONOTONE OPERATORS From ideals to operators From operators to ideals Parameterizing L*-ideals Representations of automorphism groups Applications to enlargibility Notes HOMOGENEOUS SPACES OF PSEUDO-RESTRICTED GROUPS Pseudo-restricted algebras and groups Complex polarizations Kahler polarizations Admissible pairs of operator ideals Some Kahler homogeneous spaces Notes APPENDICES Differential Calculus and Smooth Manifolds Basic Differential Equations of Lie Theory Topological Groups References Index
TL;DR: In this paper, the authors construct algebras of pseudodifferential operators on a continuous family groupoid, which was shown not to be possible by the first author in an earlier paper.
Abstract: We construct algebras of pseudodifferential operators on a continuous family groupoid , this was shown not to be possible by the first author in an earlier paper. AMS 2000 Mathematics subject classification: Primary 35S05. Secondary 35J15; 47G30; 58J40; 46L87
TL;DR: In this paper, a generalisation of the Basu-Harvey equation is presented that describes membranes ending on intersecting five-brane configurations corresponding to various calibrated geometries.
TL;DR: In this article, an extension of the definition of vertex algebras in arbitrary space-time dimensions together with their basic structure theory is proposed, and a one-to-one correspondence between these vertex algesbras and axiomatic quantum field theory with global conformal invariance (GCI) is constructed.
Abstract: We propose an extension of the definition of vertex algebras in arbitrary space–time dimensions together with their basic structure theory. A one–to–one correspondence between these vertex algebras and axiomatic quantum field theory (QFT) with global conformal invariance (GCI) is constructed.
TL;DR: In this article, a duality property for star products is established and a non-commutative algebra of operator symbols which are positive definite probability distributions is found. And the kernel of the star product is established in explicit form and examples are considered.