Showing papers on "Operator algebra published in 2002"

Simulating physical phenomena by quantum networks

Abstract: Physical systems, characterized by an ensemble of interacting constituents, can be represented and studied by different algebras of operators (observables). For example, a fully polarized electronic system can be studied by means of the algebra generated by the usual fermionic creation and annihilation operators or by the algebra of Pauli (spin-1/2) operators. The Jordan-Wigner isomorphism gives the correspondence between the two algebras. As we previously noted, similar isomorphisms enable one to represent any physical system in a quantum computer. In this paper we evolve and exploit this fundamental observation to simulate generic physical phenomena by quantum networks. We give quantum circuits useful for the efficient evaluation of the physical properties (e.g., the spectrum of observables or relevant correlation functions) of an arbitrary system with Hamiltonian H.

Classification of Nuclear C*-Algebras. Entropy in Operator Algebras

Abstract: I. Classification of Nuclear, Simple C*-algebras.- II. A Survey of Noncommutative Dynamical Entropy.

Modular localization and wigner particles

Abstract: We propose a framework for the free field construction of algebras of local observables which uses as an input the Bisognano–Wichmann relations and a representation of the Poincare group on the one-particle Hilbert space. The abstract real Hilbert subspace version of the Tomita–Takesaki theory enables us to bypass some limitations of the Wigner formalism by introducing an intrinsic spacetime localization. Our approach works also for continuous spin representations to which we associate a net of von Neumann algebras on spacelike cones with the Reeh–Schlieder property. The positivity of the energy in the representation turns out to be equivalent to the isotony of the net, in the spirit of Borchers theorem. Our procedure extends to other spacetimes homogeneous under a group of geometric transformations as in the case of conformal symmetries and of de Sitter spacetime.

Partial *- Algebras and Their Operator Realizations

Abstract: Foreword. Introduction. I: Theory of Partial O*-Algebras. 1. Unbounded Linear Operators in Hilbert Spaces. 2. Partial O*-Algebras. 3.Commutative Partial O*-Algebras. 4. Topologies on Partial O*-Algebras. 5. Tomita Takesaki Theory in Partial O*-Algebras. II: Theory of Partial *-Algebras. 6. Partial *-Algebras. 7. *-Representations of Partial *-Algebras. 8. Well-behaved X>*-Representations. 9. Biweights on Partial *-Algebras. 10. Quasi *-Algebras of Operators in Rigged Hilbert Spaces. 11. Physical Applications. Outcome. Bibliography. Index.

A Short Course on Spectral Theory

Abstract: Spectral theory and Banach algebras * Operators on Hilbert space * Asymptotics: compact perturbations and Fredholm theory * Methods and applications * Bibliography * Index

Finite Quantum Groupoids and Their Applications

Abstract: We give a survey of the theory of finite quantum groupoids (weak Hopf algebras), including foundations of the theory and applications to finite depth subfactors, dynamical deformations of quantum groups, and invariants of knots and 3-manifolds.

Three-Point Functions for MN/SN Orbifolds¶with?= 4 Supersymmetry

Abstract: The D1–D5 system is believed to have an “orbifold point” in its moduli space where its low energy theory is a ?=4 supersymmetric sigma model with target space MN/SN, where M is T4 or K3. We study correlation functions of chiral operators in CFTs arising from such a theory. We construct a basic class of chiral operators from twist fields of the symmetric group and the generators of the superconformal algebra. We find explicitly the 3-point functions for these chiral fields at large N; these expressions are “universal” in that they are independent of the choice of M. We observe that the result is a significantly simpler expression than the corresponding expression for the bosonic theory based on the same orbifold target space.

Discrete product systems of Hilbert bimodules

Abstract: A Hilbert bimodule is a right Hilbert module X over a C*-algebra A together with a left action of A as adjointable operators on X. We consider families X = {X s : s E P} of Hilbert bimodules, indexed by a semigroup P, which are endowed with a multiplication which implements isomorphisms X s ⊗ A X t → X st ; such a family is a called a product system. We define a generalized Cuntz-Pimsner algebra O x , and we show that every twisted crossed product of A by P can be realized as O x for a suitable product system X. Assuming P is quasi-lattice ordered in the sense of Nica, we analyze a certain Toeplitz extension T cov (X) of O X by embedding it in a crossed product B P A T,X P which has been twisted by X; our main Theorem is a characterization of the faithful representations of B P X T,X P.

Group C*-algebras as compact quantum metric spaces

Abstract: Let l be a length function on a group G, and let M l denote the operator of pointwise multiplication by l on l 2 (G). Following Connes, M l can be used as a Dirac operator for C* r (G). It defines a Lipschitz seminorm on C* r (G), which defines a metric on the state space of C* r (G). We investigate whether the topology from this metric coincides with the weak-* topology (our definition of a compact quantum metric space). We give an affirmative answer for G = Z d when l is a word-length, or the restriction to Z d of a norm on R d . This works for C* r (G) twisted by a 2-cocycle, and thus for non-commutative tori. Our approach involves Connes' cosphere algebra, and an interesting compactification of metric spaces which is closely related to geodesic rays.

Differential equations and intertwining operators

Abstract: We show that if every module W for a vertex operator algebra V satisfies the condition that the dimension of W/C_1(W) is less than infinity, where C_1(W) is the subspace of W spanned by elements of the form u_{-1}w for u in V of positive weight and w in 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 reducibility 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.

Observables of String Field Theory

Abstract: We study gauge invariant operators of open string field theory and find a precise correspondence with on-shell closed strings. We provide a detailed proof of the gauge invariance of the operators and a heuristic interpretation of their correlation functions in terms of on-shell scattering amplitudes of closed strings. We also comment on the implications of these operators to vacuum string field theory.

Rational Vertex Operator Algebras and the Effective Central Charge

Abstract: We establish that the Lie algebra of weight one states in a (strongly) rational vertex operator algebra is reductive, and that its Lie rank is bounded above by the effective central charge. We show that lattice vertex operator algebras may be characterized by the equalities of the effective central charge, the Lie rank and the central charge, and in particular holomorphic lattice theories may be characterized among all holomorphic vertex operator algebras by the equality of the Lie rank and the central charge.

Spectral flow and Dixmier traces

Abstract: We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the heat operator in a general semi-finite von Neumann algebra. Our results have several applications. We deduce a formula for the Chern character of an odd ${\mathcal L}^{(1,\infty)}$-summable Breuer-Fredholm module in terms of a Hochschild 1-cycle. We explain how to derive a Wodzicki residue for pseudo-differential operators along the orbits of an ergodic $\IR^n$ action on a compact space $X$. Finally we give a short proof an index theorem of Lesch for generalised Toeplitz operators.

Wild topology, hyperbolic geometry and fusion algebra of high energy particle physics

Abstract: The relation between Wild Topology, Hyperbolic Geometry and Fusion Algebra on the one side and the charge and coupling constants of the standard model and quantum gravity on the other is examined. The close connection found between E (∞) theory and the Topological theory of four manifolds as well as the theory of fundamental groups is elucidated using various classical theories and recent results due to Antoine, Wada, Alexander, Klein, Kummer, Freedman, Kaufmann, Witten, Jones and Connes.

Modular Categories and Orbifold Models

Abstract: In this paper, we try to answer the following question: given a modular tensor category ? with an action of a compact group G, is it possible to describe in a suitable sense the “quotient” category ?/G? We give a full answer in the case when ?=?ℯ? is the category of vector spaces; in this case, ?ℯ?/G turns out to be the category of representation of Drinfeld's double D(G). This should be considered as the category theory analog of the topological identity {pt}/G=BG. This implies a conjecture of Dijkgraaf, Vafa, E. Verlinde and H. Verlinde regarding so-called orbifold conformal field theories: if ? is a vertex operator algebra which has a unique irreducible module, ? itself, and G is a compact group of automorphisms of ?, and some not too restrictive technical conditions are satisfied, then G is finite, and the category of representations of the algebra of invariants, ? G , is equivalent as a tensor category to the category of representations of Drinfeld's double D(G). We also get some partial results in the non-holomorphic case, i.e. when ? has more than one simple module.

Twisted sectors for tensor product vertex operator algebras associated to permutation groups

Abstract: Let V be a vertex operator algebra, and for k a positive integer, let g be a k-cycle permutation of the vertex operator algebra V ⊗ k . We prove that the categories of weak, weak admissible and ordinary g-twisted modules for the tensor product vertex operator algebra V ⊗ k are isomorphic to the categories of weak, weak admissible and ordinary V-modules, respectively. The main result is an explicit construction of the weak g-twisted V ⊗ k -modules from weak V-modules. For an arbitrary permutation automorphism g of V ⊗ k the category of weak admissible g-twisted modules for V ⊗ k is semisimple and the simple objects are determined if V is rational. In addition, we extend these results to the more general setting of γg-twisted V ⊗ k -modules for γ a general automorphism of V acting diagonally on V ⊗ k and g a permutation automorphism of V ⊗ k .

Rationality, Regularity, and C_2-cofiniteness

Abstract: We demonstrate that, for CFT vertex operator algebras, C_2-cofiniteness and rationality is equivalent to regularity. In addition, we show that, for C_2-cofinite vertex operators algebras, irreducible weak modules are ordinary modules and C_2-cofinite, and V_L^+ are C_2-cofinite.

Weak amenability of module extensions of banach algebras

Abstract: We start by discussing general necessary and sufficient conditions for a module extension Banach algebra to be n-weakly amenable, for n = 0,1,2; Then we investigate various special cases All these case studies finally provide us with a way to construct an example of a weakly amenable Banach algebra which is not 3-weakly amenable This answers an open question raised by H G Dales, F Ghahramani and N Gronbaek

Noncommutative gauge theory without Lorentz violation

Abstract: The most popular noncommutative field theories are characterized by a matrix parameter ${\ensuremath{\theta}}^{\ensuremath{\mu}\ensuremath{ u}}$ that violates Lorentz invariance. We consider the simplest algebra in which the $\ensuremath{\theta}$ parameter is promoted to an operator and Lorentz invariance is preserved. This algebra arises through the contraction of a larger one for which explicit representations are already known. We formulate a star product and construct the gauge-invariant Lagrangian for Lorentz-conserving noncommutative QED. Three-photon vertices are absent in the theory, while a four-photon coupling exists and leads to a distinctive phenomenology.

On odd Laplace operators

Abstract: We consider odd Laplace operators acting on densities of various weights on an odd Poisson (= Schouten) manifold M. We prove that the case of densities of weight 1/2 (half-densities) is distinguished by the existence of a unique odd Laplace operator depending only on a point of an 'orbit space' of volume forms. This includes earlier results for the odd symplectic case, where there is a canonical odd Laplacian on half-densities. The space of volume forms on M is partitioned into orbits by the action of a natural groupoid whose arrows correspond to the solutions of the quantum Batalin–Vilkovisky equations. We compare this situation with that of Riemannian and even Poisson manifolds. In particular, we show that the square of an odd Laplace operator is a Poisson vector field defining an analog of Weinstein's 'modular class'.

Operators with symbol hierarchies and iterated asymptotics

Abstract: Ellipticity of (pseudo-differential) operators on a manifold with geometric singularities gives rise to a hierarchy of symbols, associated with the system of lower-dimensional strata of the configuration. Classical examples are boundary value problems with interior and boundary symbols (the latter ones describe Shapiro-Lopatinskij ellipticity of boundary conditions), or operators on manifolds with conical singularities with interior and conormal symbols. Ellipticity on a manifold with smooth edges may be investigated by a suitable combination of ideas from boundary value problems and cone calculus. The present article studies another typical case, namely ellipticity on a manifold that has edges with conical singularities. Locally, we may talk about cones, where the base is a manifold with smooth edges. Parametrices and iterated asymptotics of solutions to elliptic equations are determined by a three-component symbolic hierarchy, with interior, edge and conormal symbols. We construct an operator algebra of 2 × 2-block matrices, where the upper left corners contain the interior operators, together with so-called Green and Mellin operators (caused by analogues of Green's function in boundary value problems as well as by asymptotic phenomena), while the other entries contain trace and potential conditions with respect to the edge and pseudo-differential operators on the edge itself that are of Fuchs type with respect to the conical points. The calculus is organized in an iterative way and can be viewed as a starting point for constructing similar operator algebras with asymptotics for higher polyhedral singularities.

The su(2)_{-1/2} WZW model and the beta-gamma system

Abstract: The bosonic beta-gamma ghost system has long been used in formal constructions of conformal field theory. It has become important in its own right in the last few years, as a building block of field theory approaches to disordered systems, and as a simple representative -- due in part to its underlying su(2)_{-1/2} structure -- of non-unitary conformal field theories. We provide in this paper the first complete, physical, analysis of this beta-gamma system, and uncover a number of striking features. We show in particular that the spectrum involves an infinite number of fields with arbitrarily large negative dimensions. These fields have their origin in a twisted sector of the theory, and have a direct relationship with spectrally flowed representations in the underlying su(2)_{-1/2} theory. We discuss the spectral flow in the context of the operator algebra and fusion rules, and provide a re-interpretation of the modular invariant consistent with the spectrum.

Some Comments on the Representation Theory of the Algebra Underlying Loop Quantum Gravity

Abstract: Important characteristics of the loop approach to quantum gravity are a specific choice of the algebra A of observables and of a representation of A on a measure space over the space of generalized connections. This representation is singled out by its elegance and diffeomorphism covariance. Recently, in the context of the quest for semiclassical states, states of the theory in which the quantum gravitational field is close to some classical geometry, it was realized that it might also be worthwhile to study different representations of the algebra A of observables. The content of the present note is the observation that under some mild assumptions, the mathematical structure of representations of A can be analyzed rather effortlessly, to a certain extent: Each representation can be labeled by sets of functions and measures on the space of (generalized) connections that fulfill certain conditions. These considerations are however mostly of mathematical nature. Their physical content remains to be clarified, and physically interesting examples are yet to be constructed.

Wigner particle theory and local quantum physics

Abstract: Wigner's theory of positive energy representations of the Poincare group has often been used to give additional justifications for the Lagrangian quantization approach to QFT. Here we show that by extension with a modular localization structure it can directly lead to the net of local algebras without the use of any point-like field coordinatization. The same modular methods reveal that among the irreducible representations there are two exotic types (d = 1 + 2 massive anyons and d = 1 + 3 zero mass helicity towers) whose localization is string-like; in fact, their conversion into operator algebras leads to free string field theory. We also report on two attempts to extend the underlying spirit of the intrinsic (nonquantization) Wigner approach to the realm of interacting theories. Both aim at unravelling the structure of (Rindler) wedge-localized algebras and show, for the first time, the constructive power of the algebraic approach which, although conceived by Rudolph Haag more than 40 years ago, has primarily contributed to the structural understanding of QFT.

DT-operators and decomposability of Voiculescu's circular operator

Abstract: The DT-operators are introduced, one for every pair (\mu,c) consisting of a compactly supported Borel probability measure \mu on the complex plane and a constant c>0. These are operators on Hilbert space that are defined as limits in *-moments of certain upper triangular random matrices. The DT-operators include Voiculescu's circular operator and elliptic deformations of it, as well as the circular free Poisson operators. We show that every DT-operator is strongly decomposable. We also show that a DT-operator generates a II_1-factor, whose isomorphism class depends only on the number and sizes of atoms of \mu. Those DT-operators that are also R-diagonal are identified. For a quasi-nilpotent DT-operator T, we find the distribution of T^*T and a recursion formula for general *-moments of T.