Showing papers on "Operator algebra published in 1992"

Quantum affine algebras and holonomic difference equations

Abstract: We derive new holonomicq-difference equations for the matrix coefficients of the products of intertwining operators for quantum affine algebra representations of levelk. We study the connection opertors between the solutions with different asymptotics and show that they are given by products of elliptic theta functions. We prove that the connection operators automatically provide elliptic solutions of Yang-Baxter equations in the “face” formulation for any type of Lie algebra $$\mathfrak{g}$$ and arbitrary finite-dimensional representations of . We conjecture that these solutions of the Yang-Baxter equations cover all elliptic solutions known in the contexts of IRF models of statistical mechanics. We also conjecture that in a special limit whenq→1 these solutions degenerate again into solutions with $$q' = \exp \left( {\frac{{2\pi i}}{{k + g}}} \right)$$ . We also study the simples examples of solutions of our holonomic difference equations associated to $$U_q (\widehat{\mathfrak{s}\mathfrak{l}(2)})$$ and find their expressions in terms of basic (orq−)-hypergeometric series. In the special case of spin −1/2 representations, we demonstrate that the connection matrix yields a famous Baxter solution of the Yang-Baxter equation corresponding to the solid-on-solid model of statistical mechanics.

The CPT-theorem in two-dimensional theories of local observables

Abstract: Let ℳ be a von Neumann algebra with cyclic and separating vector Ω, and letU(a) be a continuous unitary representation ofR with positive generator and Ω as fixed point. If these unitaries induce for positive arguments endomorphisms of ℳ then the modular group act as dilatations on the group of unitaries. Using this it will be shown that every theory of local observables in two dimensions, which is covariant under translation only, can be imbedded into a theory of local observables covariant under the whole Poincare group. This theory is also covariant under the CPT-transformation.

The dual of the Haagerup tensor product

Abstract: The weak*-Haagerup tensor product Jt ®w.hjV of two von Neumann algebras is related to the Haagerup tensor product M ®h Jf in the same way that the von Neumann algebra tensor product is related to the spatial tensor product. Many of the fundamental theorems about completely bounded multilinear maps may be deduced from elementary properties of the weak*-Haagerup tensor product. We show that X* w.h Y* = (X®h Y)* for all operator spaces A'and Y. The weak*-Haagerup tensor product has simple characterizations and behaviour with reference to slice map properties. The tensor product of two (not necessarily self-adjoint) operator algebras is proven to have many strong commutant properties. All operator spaces possess a certain approximation property which is related to this tensor product. The connection between bimodule maps and commutants is explored.

Algebraic and Diagrammatic Methods in Many-Fermion Theory

Abstract: This book contains the proceedings of Algebraic and Diagrommatic Methods in Mang-Fermion Theory. Topics covered include: Operator Algebra; The Independent-Particle Model; Occupation-Number Formalism; Diagrams; The Configuration-Interaction Method; Many-Body Perturbation Theory; and The Coupled-Cluster Method.

The metric aspect of noncommutative geometry

Abstract: Most of the previous work on “noncommutative geometry” could more accurately be labeled as noncommutative differential topology, in that it deals with the homology of differential forms on noncommutative spaces (cyclic homology) and vector bundles on noncommutative spaces (K-theory) [Col]. However, the essence of geometry has to do with the metric properties of spaces.

Spatial Operator Algebra for multibody system dynamics

Abstract: The Spatial Operator Algebra framework for the dynamics of general multibody systems is described. The use of a spatial operator-based methodology permits the formulation of the dynamical equations of motion of multibody systems in a concise and systematic way. The dynamical equations of progressively more complex grid multibody systems are developed in an evolutionary manner beginning with a serial chain system, followed by a tree topology system and finally, systems with arbitrary closed loops. Operator factorizations and identities are used to develop novel recursive algorithms for the forward dynamics of systems with closed loops. Extensions required to deal with flexible elements are also discussed.

Facial Structure in Operator Algebra Theory

Abstract: We give a complete description of the closed faces in twelve kinds of convex sets that appear in operator algebra theory. These consist of positive parts of unit balls for C*-algebras and their dual spaces, and for von Neumann algebras and their pre-duals; of self-adjoint parts of unit balls in the same four classes and finally of general unit balls in the four classes. All these faces are shown to be semi-exposed and naturally paired with a polar face in the dual (or pre-dual) space

On some algebraic structure arising in string theory

Abstract: Lian and Zuckerman proved that the homology of a topological chiral algebra can be equipped with the structure of a BV-algebra; \ie one can introduce a multiplication, an odd bracket, and an odd operator $\Delta$ having the same properties as the corresponding operations in Batalin-Vilkovisky quantization procedure. We give a simple proof of their results and discuss a generalization of these results to the non chiral case. To simplify our proofs we use the following theorem giving a characterization of a BV-algebra in terms of multiplication and an operator $\Delta$: {\em If $A$ is a supercommutative, associative algebra and $\Delta$ is an odd second order derivation on $A$ satisfying $\Delta^2=0$, one can provide $A$ with the structure of a BV-algebra.}

Phase transitions and algebra of fluctuation operators in an exactly soluble model of a quantum anharmonic crystal

Abstract: A complete description of the fluctuation operator algebra is given for a quantum crystal showing displacement structural phase transitions. In the one-phase region, the fluctuations are normal and its algebra is non-Abelian. In the two-phase region and on the critical line (T c >0) the momentum fluctuation is normal, the displacement is critical, and the algebra is Abelian; atT c =0 (quantum phase transition) this algebra is non-Abelian with abnormal displacement and supernormal (squeezed) momentum fluctuation operators, both being dimension dependent.

Causal Nets of Operator Algebras: Mathematical Aspects of Algebraic Quantum Field Theory

Abstract: Fundamentals of operator algebras causal nets of operator algebras superselection theory causal nets of algebras and quantum fields modular theory, type theory and injectivity of local algebras.

Nambu mechanics and its quantization

Abstract: The algebra of observables inherent in the Nambu formalism [Phys. Rev. D 7, 2405 (1973)] for a generalization of classical Hamiltonian dynamics is investigated. A consistency requirement of time evolution of the Nambu bracket leads to a five-point identity. Two types of algebras are possible at the classical level. Their composition properties under a tensor product are considered and the physical implications are analyzed. A quantum generalization of these algebras is shown to be impossible.

Gravitons as embroidery on the weave

Abstract: We investigate the physical interpretation of the loop states that appear in the loop representation of quantum gravity. By utilizing the “weave” state, which has been recently introduced as a quantum description of the microstructure of flat space, we analyze the relation between loop states and graviton states. This relation determines a linear map M from the state-space of the nonperturbative theory (loop space) into the state-space of the linearized theory (Fock space). We present an explicit form of this map, and a preliminary investigation of its properties. The existence of such a map indicates that the full nonperturbative quantum theory includes a sector that describes the same physics as (the low energy regimes of) the linearized theory, namely gravitons on flat space.

Analysis of the superconformal cohomology structure of N=4 super Yang-Mills

Abstract: The nilpotent symmetry operator for the full superconformal symmetry of N=4 super Yang-Mills is constructed using the method of Batalin and Vilkovisky (1977). Spectral sequences are then used to prove that the cohomology of this operator is trivial, and so the theory has no possible anomalies. This shows that all symmetries are preserved after quantization, and so there is no conformal anomaly and the theory is thus finite to all orders.

The quantum strip: Liouville theory for open strings

Abstract: The quantum group structure of 2D gravity recently put forward by one of us (J.-L. G.) is used to study quantum gravity on the strip. The boundary conditions, previously studied by A. Neveu and this author become easy to implement when one introduces the universal family of chiral operators associated withUq(sl(2)). A general formula for inverse powers of the metric-tensor operator is thereby derived. It contains a new universal matrixA, acting in representation-space, which obeys identities involving theR matrix, the Clebsch-Gordon coefficients, and the co-products ofUq(sl(2)). The physical meaning of these identities is to ensure that these powers of the metric are local and closed by fusion.

Algebraic Aspects in Modular Theory

Abstract: Nowadays, the so called modular theory for von Neumann algebras is well-established and fully utilized (see [17] for example) in the field of operator algebras. The symbols conventionally used there, however, does not seem to be so much expressive in some sense. One of the main purposes in the present article is a focussed account of the problem of this kind for modular theory in operator algebras. In the past time, there had been already some suggestions on the improvement of notations for the modular theory but not in a thorough way. Among them, Woronowicz's approach [22] and new symbols introduced in [2] are worthy of attention. Around the same time of these works, the non-commutative //-theory for arbitrary von Neumann algebras came out and had been developped by several people such as Haagerup, Connes-Hilsum, Kosaki, and Araki-Masuda, and so on. It is worth pointing out the fact that, in this theory, the l/p-th power of a state of a von Neumann algebra is identified with an element in the relevant //-space. Now we give a brief outline of the contents in this article. The first section surveys the background materials and the substantial parts start from the next section.

A comment on a recent work of Borchers

Abstract: We establish a converse of a recent result of Borchers

Dense m-convex Frechet Subalgebras of Operator Algebra Crossed Products by Lie Groups

Abstract: Let A be a dense Frechet *-subalgebra of a C*-algebra B. (We do not require Frechet algebras to be m-convex.) Let G be a Lie group, not necessarily con- nected, which acts on both $A$ and B by *-automorphisms, and let \s be a sub- multiplicative function from G to the nonnegative real numbers. If \s and the action of G on A satisfy certain simple properties, we define a dense Frechet *-subalgebra G\rtimes^{\s} A of the crossed product L^{1}(G, B). Our algebra consists of differentiable A-valued functions on G, rapidly vanishing in \s. We give conditions on the action of G on A which imply the m-convexity of the dense subalgebra G\rtimes^{\s}A. A locally convex algebra is said to be m-con- vex if there is a family of submultiplicative seminorms for the topology of the algebra. The property of m-convexity is important for a Frechet algebra, and is useful in modern operator theory. If G acts as a transformation group on a manifold M, we develop a class of dense subalgebras for the crossed product L^{1}(G, C_{0}(M)), where C_{0}(M) denotes the continuous functions on M vanishing at infinity with the sup norm topology.We define Schwartz functions S(M) on M, which are differentiable with respect to some group action on M, and are rapidly vanishing with respect to some scale on M. We then form a dense m-convex Frechet *-subalgebra G\rtimes^ {\s} S(M) of rapidly vanishing, G-differentiable functions from G to S(M). If the reciprocal of \s is in L^{p}(G) for some p, we prove that our group algebras S^{\s}(G) are nuclear Frechet spaces, and that G\rtimes^{\s}A is the projective completion S^{\s}(G) \otimes A.

Symmetry and quantization: Higher-order polarization and anomalies

Abstract: The concept of (full) polarization subalgebra in a Group Approach to Quantization on a Lie group G as a generalization of the analogous concept in geometric or standard quantization is discussed. The lack of full polarization subalgebras is considered as an anomaly of the corresponding system and related to its more conventional definition. A generalization of the subalgebra of (full) polarization is then provided, made out of higher‐order differential operators in the enveloping algebra of G. Higher‐order polarizations can also be used to quantize nonanomalous theories in different ‘‘representations.’’ Numerous examples are analyzed, including the finite‐dimensional dynamics associated with the Schrodinger group, which presents an anomaly, and an infinite‐dimensional anomalous system associated with the Virasoro group. In the last example, the operators in the higher‐order polarization are in one‐to‐one correspondence with the null vectors in the Verma module approach.

Recursive formulation of operational space control

Abstract: A recently developed spatial operator algebra approach to modeling and analysis of multibody robotic systems is used to develop O(n) recursive algorithms that compute the op erational space mass matrix and the operational space coriolis/centrifugal and gravity terms of an n-link serial manipulator. These algorithms enable an O(n) recursive im plementation of operational space control.

From quantum fields to local von neumann algebras

Abstract: The subject of the paper is an old problem of the general theory of quantized fields: When can the unbounded operators of a Wightman field theory be associated with local algebras of bounded operators in the sense of Haag? The paper reviews and extends previous work on this question, stressing its connections with a noncommutive generalization of the classical Hamburger moment problem. Necessary and sufficient conditions for the existence of a local net of von Neumann algebras corresponding to a given Wightman field are formulated in terms of strengthened versions of the usual positivity property of Wightman functionals. The possibility that the local net has to be defined in an enlarged Hilbert space cannot be ruled out in general. Under additional hypotheses, e.g., if the field operators obey certain energy bounds, such an extension of the Hilbert space is not necessary, however. In these cases a fairly simple condition for the existence of a local net can be given involving the concept of “central positivity” introduced by Powers. The analysis presented here applies to translationally covariant fields with an arbitrary number of components, whereas Lorentz covariance is not needed. The paper contains also a brief discussion of an approach to noncommutative moment problems due to Dubois-Violette, and concludes with some remarks on modular theory for algebras of unbounded operators.

Nuclear *-algebras have amenable unitary groups

Abstract: Let A be a unital C*-algebra with unitary group G Give G the relative (Banach space) weak topology Then G is a topological group, and we show that A is nuclear if and only if there exists a left invariant mean on the space of right uniformly continuous, bounded, complex-valued functions on G Nuclear C*-algebras and invective von Neumann algebras are of fundamental importance in operator algebra theory The two classes of algebras are related through the remarkable result (of Choi-Effros and Connes): a C*-algebra A is nuclear if and only if A** is an invective von Neumann algebra From the work of Haagerup [3], nuclearity is the same as amenability (in the Banach algebra sense) for C*-algebras Further, from the deep work of Connes and others, it is known that injectivity, Property P, hyperfiniteness and amenability are all equivalent for a von Neumann algebra M The relationship between injectivity and classical amenability for topological groups is established in a result of de la Harpe [4] discussed below We note here that Haagerup in [3] proves that the injectivity of M is equivalent to the existence of a left invariant mean on a certain space of functions on the isometry semigroup of M The author plans to discuss the relationship between the invariant mean results of Haagerup and de la Harpe in a future paper We recall some notions from topological group theory A fundamental system of entourages for the right uniformity on a topological group G is given by sets of the form {(x,y) E Gx G:yx-1 E V}, where V is a neighborhood of the identity e in G Let RUC(G) be the space of right uniformly continuous bounded functions f: G -k C It is well known and easy to show that if f: G -C is bounded, then f E RUC(G) if and only if the map x -fx is norm continuous from G to loo(G), where fx(y) = f(xy) (y E G) The space RUC(G) is a unital C*-subalgebra of I, (G),and is right invariant in the sense that fx E RUC(G) whenever f E RUC(G) If X is a right invariant, unital subspace of lo (G), then an element m E X* is called a left invariant mean if m(1) = 1 = Ilmll and m(fx) = m(f) for all f E X, x E G Let Y(X) be the set of left invariant means on X Received by the editors September 10, 1990 1980 Mathematics Subject Classification (1985 Revision) Primary 46L05 ? 1992 American Mathematical Society 0002-9939/92 $100 + $25 per page

Field operators for anyons and plektons

Abstract: Given its superselection sectors with non-abelian braid group statistics, we extend the algebraA of local observables into an algebra ℱ containing localized intertwiner fields which carry the superselection charges. The construction of the inner degrees of freedom, as well as the study of their transformation properties (quantum symmetry), are entirely in terms of the superselection structure of the observables. As a novel and characteristic feature for braid group statistics, Clebsch-Gordan and commutation “coefficients” generically take values in the algebra ℳ of symmetry operators, much as it is the case with quasi-Hopf symmetry.A, ℱ, and ℳ are allC* algebras, i.e. represented by bounded operators on a Hilbert space with positive metric.

Improper filtrations for C*-algebras: spectra of unilateral tridiagonal operators

Abstract: We extend the results of our previous paper "C*-algebras and numerical linear algebra" to cover the case of "unilateral" sections. This situation bears a close resemblance to the case of Toeplitz operators on Hardy spaces, in spite of the fact that the operators here are far from Toeplitz operators. In particular, there is a short exact sequence 0 --> K --> A --> B --> 0 whose properties are essential to the problem of computing the spectra of self adjoint operators.

Commutative Banach algebras and decomposable operators

Abstract: For a multiplication operator on a semi-simple commutative Banach algebra, it is shown that the decomposability in the sense of Foias is equivalent to weak and to super-decomposability. Moreover, it can also be characterized by a convenient continuity condition for the Gelfand transform on the spectrum of the underlying Banach algebra. This result implies various permanence properties for decomposable multiplication operators and leads also to a useful characterization of the regularity for a semi-simple commutative Banach algebra. Finally, the greatest regular closed subalgebra of a commutative Banach algebra is investigated, and some applications to decomposable convolution operators on locally compact abelian groups are given.