Showing papers on "Operator algebra published in 1990"

C*-Algebras and Operator Theory

Abstract: Elementary Spectral Theory C*-Algebras and Hilbert Space Operators Ideals and Positive Functionals Von Neumann Algebras Representations of C*-Algebras Direct Limits and Tensor Products K-Theory of C*-Algebras

Braid statistics in local quantum theory

Abstract: We present details of a mathematical theory of superselection sectors and their statistics in local quantum theory over (two- and) three-dimensional space-time. The framework for our analysis is algebraic quantum field theory. Statistics of superselection sectors in three-dimensional local quantum theory with charges not localizable in bounded space-time regions and in two-dimensional chiral theories is described in terms of unitary representations of the braid groups generated by certain Yang-Baxter matrices. We describe the beginnings of a systematic classification of those representations. Our analysis makes contact with the classification theory of subfactors initiated by Jones. We prove a general theorem on the connection between spin and statistics in theories with braid statistics. We also show that every theory with braid statistics gives rise to a “Verlinde algebra”. It determines a projective representation of SL(2, ℤ) and, presumably, of the mapping class group of any Riemann surface, even if t...

Tensor representation of the quantum group SL q (2,C) and quantum Minkowski space

Abstract: We investigate the structure of the tensor product representation of the quantum groupSLq(2,C) by using the 2-dimensional quantum plane as a building block. Two types of 4-dimensional spaces are constructed applying the methods used in twistor theory. We show that the 4-dimensional real representation ofSLq(2,C) generates a consistent non-commutative algebra, and thus it provides a quantum deformation of Minkowski space. The transformation of this 4-dimensional space gives the quantum Lorentz groupSOq(3, 1).

The quantum group structure of 2D gravity and minimal models I

Abstract: On the unit circle, an infinite family of chiral operators is constructed, whose exchange algebra is given by the universalR-matrix of the quantum groupSL(2) q . This establishes the precise connection between the chiral algebra of two dimensional gravity or minimal models and this quantum group. The method is to relate the monodromy properties of the operator differential equations satisfied by the generalized vertex operators with the exchange algebra ofSL(2) q . The formulae so derived, which generalize an earlier particular case worked out by Babelon, are remarkably compact and may be entirely written in terms of “q-deformed” factorials and binomial coefficients.

A property of purely infinite simple C^*-algebras

Abstract: An alternative proof is given for the fact ([ 13]) that a purely infinite, simple C *-algebra has the FS property: the set of self-adjoint elements with finite spectrum is norm dense in the set of all self-adjoint elements. In particular, the Cuntz algebras O, (2 < n < +oo) and the Cuntz-Krieger algebras 0A if A is an irreducible matrix, have the FS property. This answers a question raised in [2, 2.10] concerning the structure of projections in the Cuntz algebras. Moreover, many corona algebras and multiplier algebras have the FS property. A C -algebra A is said to be purely infinite if (xAx) contains an infinite projection for every nonzero positive element x in A ([7, 12]). The author recently proved ([13]) that purely infinite, simple C*-algebras have the FS property; namely, the set of self-adjoint elements with finite spectrum is norm dense in the set of all self-adjoint elements. Actually, many interesting C*-algebras have the FS property. For example, the Bunce-Deddens algebras have FS ([1, 3]); many corona algebras and multiplier algebras have FS ([5, 13]); certain irrational rotation algebras have FS ([6]). Certainly, all AF algebras, von Neumann algebras, and A W* algebras have FS. In this short note, we provide another proof for the fact that purely infinite, simple C*-algebras have the FS property. The algebras ON (2 < n < +00) and OA, if A is an irreducible matrix, are purely infinite and simple ([7, 8, 9]), and many corona algebras are purely infinite and simple ([12, 13]). Hence, these C*-algebras have the FS property. In particular, this answers a question of B. Blackadar raised in [2, 2.10] concerning the projection structure of the Cuntz algebras. 1. Theorem. If A is a purely infinite, simple C* -algebra, then A has the FS property, and hence RR(A) = 0. Proof. To prove the conclusion, by [2, 2.7; 10], it is equivalent to prove that every hereditary C*-subalgebra of A has an approximate identity consisting of Received by the editors April 17, 1989 and, in revised form, September 5, 1989; the results in this paper were presented at the 17th Annual Canadian Symposium on Operator Algebras and Operator Theory, University of Toronto, May 22-26, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 46L05.

The structure of the W ∞ algebra

Abstract: We prove rigorously that the structure constants of the leading (highest spin) linear terms in the commutation relations of the conformal chiral operator algebraW∞ are identical to those of the Diff0+ ℝ2 algebra generated by area preserving diffeomorphisms of the plane. Moreover, all quadratic terms of theWN algebra are found to be absent in the limitN→∞. In particular we show thatW∞ is a central extension of Diff0+ ℝ2 with non-trivial cocycles appearing only in the commutation relations of its Virasoro subalgebra. We also propose a representation ofW∞ in terms of a single scalar field in 2+1 dimensions and discuss its significance in the context of quantum field theory.

Conformal field algebras with quantum symmetry from the theory of superselection sectors

Abstract: According to the theory of superselection sectors of Doplicher, Haag, and Roberts, field operators which make transitions between different superselection sectors—i.e. different irreducible representations of the observable algebra—are to be constructed by adjoining localized endomorphisms to the algebra of local observables. We find the relevant endomorphisms of the chiral algebra of observables in the minimal conformal model with central chargec=1/2 (Ising model). We show by explicit and elementary construction how they determine a representation of the braid groupB∞ which is associated with a Temperley-Lieb-Jones algebra. We recover fusion rules, and compute the quantum dimensions of the superselection sectors. We exhibit a field algebra which is quantum group covariant and acts in the Hilbert space of physical states. It obeys local braid relations in an appropriate weak sense.

Introduction to quantum groups

Abstract: This mini-course presents an approach to quantum groups based on the quantization of Poisson-Lie groups. In this connection the Quantum Yang-Baxter Equation and the algebraic definition of quantum groups appear quite naturally. We discuss quantum groups corresponding to simple Lie groups of classical type, their quantum vector spaces and quantum universal enveloping algebras. In particular the latter are introduced as dual objects to quantum groups with the duality given by a quantum R-matrix.

Quantum norm theory and the quantisation of metric topology

Abstract: One approach to building a genuine theory of quantum topology would be to construct a quantum theory on the set M(X) of all metrics on a set X. The authors move towards this goal by showing that, for a finite set X, almost all such metrics can be obtained by embedding X into a vector space V and then varying the norm on V. This leads to the subject of 'quantum norm theory' and they give an explicit Fock space representation of such a system. They discuss a model Hamiltonian which can produce a change in metric topology by changing the effective number of points in X.

Grassmannian coset models and unitary representations of w

Abstract: It is shown that the 2 – d coset models SU(p + 1)N/SU(p)N ⊗ U(1) provide unitary representations of the chiral operator algebra W∞ in the large level (N → ∞) limit, with central charge c = 2p. For p ≥ 2, the corresponding field theories possess additional symmetries which given rise to a U(p) matrix generalization of W∞, denoted by $W_\infty^p$. Its commutation relations are obtained in closed form for all values of p and W∞ is identified with the U(1) trace part of $W_\infty^p$. It is also shown that $W_\infty^p$ at large p is associated with the algebra of symplectic diffeomorphisms in four dimensions.

The construction of the q-analogues of the harmonic oscillator operators from ordinary oscillator operators

Abstract: The q-analogues of boson operators are constructed from ordinary boson operators by an embedding approach. Accordingly the q-deformed quantum algebras SU(n)q can be obtained straightforwardly from corresponding universal enveloping algebras of Lie algebra SU(n).

Z / nz conformal field theories

Abstract: In this paper, we give a brief introduction to Conformal Field Theory (CFT) following the presentation of G Segal We explain how to reconstruct part of a CFT from its fusion rules The possible choices of S matrices are indexed by some automorphisms of the fusion algebra We illustrate this procedure by computing the modular properties of the possible genus-one characters when the fusion algebra is the representation algebra of a finite group We also classify the modular invariant partition functions of these theories We recover as special cases the A N (1) WZW theories and the rational gaussian model

Non-Factorizable Separable Systems and Higher-Order Symmetries of the Dirac Operator

Abstract: It is shown that there exist separable systems for the Dirac operator on four-dimensional lorentzian spin manifolds that are not factorizable in the sense of Miller. The symmetry operators associated to these new separable systems are of higher order than the Dirac operator. They are characterized in the second-order case in terms of quadratic first integrals of the geodesic flow satisfying additional invariant conditions.

Heisenberg's picture and non commutative geometry of the semi classical limit in quantum mechanics

Abstract: We propose a new framework based upon non commutative geometry to control the semi classical limit in phase space. It lead in particular to uniform estimates of Nekhoroshev's type, as Planck's constant tends to zero, for the perturbation expansion Nous proposons un cadre nouveau fonde sur la geometrie non commutative, pour controler la limite semi-classique dans l'espace des phases. Cette approche permet d'obtenir des estimations, uniformes par rapport a la constante de Planck, sur la serie de perturbation, semblables a celles de Nekhoroshev en mecanique classique

C* Algebraic Quantization and the Origin of Topological Quantum Effects

Abstract: Concepts from the theory of abstract operator algebras are used to solve the problem of quantizing a particle moving on an arbitrary locally compact homogeneous space. Inequivalent quantizations are identified with inequivalent irreducible representations of the corresponding C *-algebra. Topological terms in the action (or Hamiltonian) are found to be representation-dependent, and are automatically induced by the quantization procedure. Known charge quantization conditions turn out to be identically satisfied. Several examples are considered, among them the Dirac monopole and the Aharonov-Bohm effect.

Affine geometric proofs of the Banach Stone theorems of Kadison and Kaup

Abstract: This approach was abandoned by Kadison because “the sparseness of knowledge concerning the pure states of an operator algebra makes this procedure seem difficult” [14, p. 326]. Instead he gives an intrinsic proof, depending mainly on spectral theory and the geometry of the underlying Hilbert spaces on which A and B act. Kadison points out that ρ preserves the quantum mechanical structure of the C∗-algebras, i.e., the linear structure and the power structure of self-adjoint elements. It follows, and this is significant for the viewpoint expressed in this paper, that T , given by (1.1), preserves powers of the form aa∗a, aa∗aa∗a, . . . , and hence, by polarization, that T preserves the triple product ab∗c+ cb∗a.

Exotic cohomology and index theory

Abstract: The main result of this paper is a version of the Atiyah-Singer index theorem for Dirac-type operators on (noncompact) complete Riemannian manifolds. The statement of the theorem involves a novel "cohomology" theory for such manifolds. This theory, called exotic cohomology, depends on the structure at infinity of a space; more precisely, it depends on the way that large bounded sets fit together. For each cohomology class in this theory, we define a "higher index" of a Dirac-type operator, enjoying the stability and vanishing properties of the usual Atiyah-Singer index; these higher indices are analogous to the Novikov higher signatures. Our main theorem will compute these higher indices in terms of standard topological invariants. Applications of this index theorem include a different approach to some of the results of Gromov and Lawson [10] on topological obstructions to positive scalar curvature. The concept of index that we will use involves the A^-theory functors K0 and Kx for operator algebras [3, 14]. Suppose that B is an ideal in a unital algebra C, and let T e C be invertible modulo B. (In the classical Atiyah-Singer index theorem, one takes C to be the bounded operators on the L space of some compact manifold, B the compact operators, and T an elliptic pseudodifferential operator of order zero.) Then T has an "index" in the ^-theory group KQ(B) (in the classical case this is just Z , and one recovers the usual Fredholm index). Now let M be a complete Riemannian manifold, possibly noncompact. In [18] I introduced an algebra %?(M) which is defined as follows: Sf(M) consists of all bounded operators A on L(M) that have a kernel representation