Showing papers on "Operator algebra published in 1986"

K-Theory for Operator Algebras

Abstract: I. Introduction To K-Theory.- 1. Survey of topological K-theory.- 2. Overview of operator K-theory.- II. Preliminaries.- 3. Local Banach algebras and inductive limits.- 4. Idempotents and equivalence.- III. K0-Theory and Order.- 5. Basi K0-theory.- 6. Order structure on K0.- 7. Theory of AF algebras.- IV. K1-Theory and Bott Periodicity.- 8. Higher K-groups.- 9. Bott Periodicity.- V. K-Theory of Crossed Products.- 10. The Pimsner-Voiculescu exact sequence and Connes' Thorn isomorphism.- 11. Equivariant K-theory.- VI. More Preliminaries.- 12. Multiplier algebras.- 13. Hilbert modules.- 14. Graded C*-algebras.- VII. Theory of Extensions.- 15. Basic theory of extensions.- 16. Brown-Douglas-Fillmore theory and other applications.- VIII. Kasparov's KK-Theory.- 17. Basic theory.- 18. Intersection product.- 19. Further structure in KK-theory.- 20. Equivariant KK-theory.- IX. Further Topics.- 21. Homology and cohomology theories on C*-algebras.- 22. Axiomatic K-theory.- 23. Universal coefficient theorems and Kunneth theorems.- 24. Survey of applications to geometry and topology.

Partially ordered abelian groups with interpolation

Abstract: A branch of ordered algebraic structures has grown, motivated by $K$-theoretic applications and mainly concerned with partially ordered abelian groups satisfying the Riesz interpolation property This monograph is the first source in which the algebraic and analytic aspects of these interpolation groups have been integrated into a coherent framework for general reference The author provides a solid foundation in the structure theory of interpolation groups and dimension groups (directed unperforated interpolation groups), with applications to ordered $K$-theory particularly in mind Although interpolation groups are defined as purely algebraic structures, their development has been strongly influenced by functional analysis This cross-cultural development has left interpolation groups somewhat estranged from both the algebraists, who may feel intimidated by compact convex sets, and the functional analysts, who may feel handicapped by the lack of scalars This book, requiring only standard first-year graduate courses in algebra and functional analysis, aims to make the subject accessible to readers from both disciplines High points of the development include the following: characterization of dimension groups as direct limits of finite products of copies of the integers; the double-dual representation of an interpolation group with order-unit via affine continuous real-valued functions on its state space; the structure of dimension groups complete with respect to the order-unit norm, as well as monotone sigma-complete dimension groups and dimension groups with countably infinite interpolation; and an introduction to the problem of classifying extensions of one dimension group by another The book also includes a development of portions of the theory of compact convex sets and Choquet simplices, and an expository discussion of various applications of interpolation group theory to rings and $C^*$-algebras via ordered $K_0$ A discussion of some open problems in interpolation groups and dimension groups concludes the book Of interest, of course, to researchers in ordered algebraic structures, the book will also be a valuable source for researchers seeking a background in interpolation groups and dimension groups for applications to such subjects as rings, operator algebras, topological Markov chains, positive polynomials, compact group actions, or other areas where ordered Grothendieck groups might be useful This is a reprint of the 1986 original (SURV/20S)

Differential geometry of quaternionic manifolds

Abstract: On developpe l'algebre necessaire pour comprendre les proprietes d'une variete quaternionique. On caracterise la structure quaternionique par l'existence de certains complexes d'operateurs differentiels

On the connection between quantum fields and von Neumann algebras of local operators

Abstract: The relationship between a standard local quantum field and a net of local von Neumann algebras is discussed. Two natural possibilities for such an association are identified, and conditions for these to obtain are found. It is shown that the local net can naturally be so chosen that it satisfies the Special Condition of Duality. The notion of an intrinsically local field operator is introduced, and it is shown that such an operator defines a local net with which the field is locally associated. A regularity condition on the field is formulated, and it is shown that if this condition holds, then there exists a unique local net with which the field is locally associated if and only if the field algebra contains at least one intrinsically local operator. Conditions under which a field and other fields in its Borchers class are associated with the same local net are found, in terms of the regularity condition mentioned.

Operator ordering in quantum mechanics and quantum gravity

Abstract: A nonperturbative approach to the quantization of the canonical algebra of pure gravity is presented. The problem of factor ordering of operators in the constraintsOpen image in new windowμΨ=0 is resolved by invoking Hermiticity under the invariant inner product in hyperspace—the space of all three-dimensional metricsgij(x)—and covariance under co-ordinate transformations. The resulting operatorsOpen image in new windowμ receive corrections of order\( / \) and\(h / ^2 \) only and the algebra closes up to a conformal anomaly term. If the algebra is enlarged by the inclusion of the anomalous operator, it can be shown that, by some suitable choice of the gauge parameter corresponding to this unphysical symmetry, the integrated form of the algebra can be made to close.

A nil algebra of bounded operators on Hilbert space with semisimple norm closure

Abstract: An algebra of operators having the property of the title is constructed and it is used to give examples related to some recent invariant subspace results.

Some Results on Cohomology with Borel Cochains, with Applications to Group Actions on Operator Algebras

Abstract: This paper is an outgrowth of joint work with Richard Herman [5] and Iain Raeburn [13]. In both of these projects, questions concerning group actions on operator algebras naturally led to a study of obstruction classes in H2(G,U(A)), where U(A) is the (suitably topologized) unitary group of an abelian operator algebra A. The appropriate cohomology theory here is the “Borel cochain” theory of C.C.Moore, as developed and systematized in [8]. In case A is a von Neumann algebra, U(A) is essentially what Moore calls U(X,T) (X here is some standard measure space), and machinery for computing the relevant cohomology groups is developed and applied in [8] and [9].

Vibrational excitation in collisions between two diatomic molecules using an operator algebra

Abstract: We introduce an operator algebra to study vibrational–translational energy transfer and apply it to collinear collisions between two diatomic molecules. The system is modeled by two linearly driven parametric oscillators with a bilinear time‐dependent coupling between them. We describe the time evolution of the linearly driven parametric oscillators accounting for part of the coupling with a sequence of transformations that reduces the coupling at each step, and use perturbation theory to account for the remainder. Results of a basis set expansion are compared with those of the algebraic approach for the collisions N2+O2, N2+CO, and H2+H2. The algebraic approach requires solving a substantially smaller number of coupled differential equations, and gives very good agreement for all systems, for several transitions and relative collision energies.

Operator algebra and correlation functions in the two-dimensional SU(2) x SU(2) chiral Wess-Zumino model

Abstract: We investigate a conformally invariant solution of the SU(2) x SU(2) chiral Wess-Zumino model in two-dimensional space--time. We determine exactly the anomalous dimensions of all fields, the structure constants of the operator algebra, and the four-point correlation functions.

Consistent algebra for the constraints of quantum gravity

Abstract: In the previous paper, a proposal was advanced for the ordering of the operatorsOpen image in new windowμ that arise in Dirac's programme for the quantization of gravity. The resulting algebra, however, was found to contain an undesired anomalous operator. Here we present a minimal modification of the canonical commutation relations of gravity in order to ensure that covariance is maintained for noncommuting tensor operators. As a result of the modification, the algebra of the quantum operator constraints is found to close exactly as in the classical case.

Multivariable Wiener-Hopf operators I. Representations

Abstract: On identifie toutes les representations de la C*-algebre des operateurs de convolution de Wiener-Hopf a plusieurs variables pour des cones convexes fermes

Analysis on generalized superspaces

Abstract: The analysis over σ‐commutative algebras (generalized supercommutative algebras), that is, differentiation and integration for functions defined on superspace over a σ‐commutative algebra, is studied.

Quantum mechanics in coherent algebra on phase space

Abstract: Quantum mechanics is formulated on a quantum mechanical phase space. The algebra of observables and states is represented by an algebra of functions on a phase space that fulfils a certain coherence condition, expressing the quantum mechanical superposition principle. The trace operation is an integration over phase space. In the case where the canonical variables independently run from - infinity to + infinity formalism reduces to the representation of quantum mechanics by Wigner distributions. However, the notion of coherent algebra allows one to apply the formalism to spaces for which the Wigner mapping is not known. The quantum mechanics of a particle in a plane in polar coordinates is discussed as an example.

Field algebras do not leave field domains invariant

Abstract: It is proved that von Neumann algebras associated to Op*-algebra (P, D) cannot leave the domainD ofP invariant if they are type I or type III factors or finite direct sums of such factors. Hence it follows that in quantum field theory global and local von Neumann field algebras in typical cases do not leave invariant the definition domain of Wightman fields.

On operator algebras and operator ranges

Abstract: On etudie si une algebre d'operateurs fortement fermee cyclique de domaine doit etre strictement cyclique. On montre que le commuttant d'une algebre cyclique de domaine a des sous-espaces invariants non triviaux

On representations of an operator algebra and the transfer matrix spectrum of the q-state Potts model

Abstract: The author shows that the Potts and Temperley-Lieb representations of the projection generators of a von Neumann algebra are, in general, reducible. The author writes down a new representation with non-vanishing product of odd generators R and gives evidence that it is a common element in these reductions. The operators in this representation may be interpreted as giving the bond transfer matrices of the square lattice Whitney polynomial. The author shows that the reduction of the Temperley-Lieb representation also contains irreducible representations with R=0 which are responsible for eigenvalues in the ice-model spectrum independent of those of the Potts model.

Counterexamples in the theory of nonselfadjoint operator algebras

Abstract: In this note we announce the answers to several questions which involve nonselfadjoint operator algebras Detailed proofs will appear elsewhere We use the following notation M is a separable Hilbert space, B(#) is the algebra of bounded linear operators on )/, and Bi(H) is the ideal of trace class operators on M For T € 0(#), {T}' is the commutant of T and {T}" is the double commutant of T B(M) is the dual of Bi(#) (see [2]) so that B{M) has a weak * topology A(T) denotes the smallest weak * closed algebra containing T and 7, while IV (T) is the smallest weak operator closed algebra containing T and I LatT is the lattice of (closed) invariant subspaces of T, and AlgLatT = {B £ B{X): LatT C La t£} It is elementary that A{T) C IV {T) C {T}" C {T}', that IV (T) C AlgLatT, and that all of these sets except A(T) are weakly closed algebras Further, T is said to be reflexive if IV (T) = Alg Lat T We will consider the following questions QUESTION 1 Does 1V{T) = {T}' n AlgLatT, VT e B{U)t QUESTION 2 Does 1V{T) = {T}" n AlgLatT, VT 1? (Here T denotes the direct sum of n copies of T) QUESTION 4 If Ti and T2 are reflexive operators, must Ti0T2 be reflexive? QUESTION 5 Does A{T) = W(T), VT e B(#)? QUESTION 6 Does IV(T) have a separating vector, VT c S(>/)? Before stating the last question, we need some additional notation Since TV(T) is weak * closed in B(#), 1V(T) is a dual space, with predual Tj/(T)* = B I ( ^ ) / W ( T ) _ L Here W(T)j denotes the preannihilator of U/(T) For each n, let jPn C Bi(#) denote the set of operators of rank < n QUESTION 7 Is Fi/U/(T)i dense in W(T)*, VT C B(#)? Some remarks regarding these questions are in order There are some relations among the questions For n = 1,2, or 6, an affirmative answer to Question n implies an affirmative answer to Question n + 1 Question 1 was raised independently by D Sarason and P Rosenthal (see [6, p 195] and [7]) Rosenthal also asked Question 2 in [7] In [4], J Deddens listed several open questions, including Questions 3 and 4, concerning reflexive operators Question 5 has been raised by many people The question appears in [2] In [8], D Westwood gave an example of an operator T so that A(T) = IV (T) but so that the weak and weak * topologies are different on A(T)

On the possible existence of modulated phases in the random chessboard model

Abstract: The random chessboard Hamiltonian describes a spin system in D = 2 dimensions with two Ising-like variables per site that interact via a next-nearest neighbor ( k 2 ) and a four-spin (Γ 0 ) coupling constants. In this work we present an improvement of the early considered renormalization group-flow equations (RGFE) for this model. We introduce an unsteady basis of six marginal operators that diagonalize the Kadanoff operator algebra coefficients with the instantaneous four-spin Hamiltonian. The equations of motion are formulated in this ‘noninertial’ frame in a standard way and afterwards they are written using a fixed frame of marginal operators. The resulting nonlinear first-order differential equation is studied. We find that after a ‘time’ l = l c , the renormalized coupling become complex numbers. For l →∞ the renormalized Hamiltonian iterates to a double eight-vertex fixed line. As the approach to this limit occurs with oscillations of ground frequency ω ≅ Γ 0 , we deduce the possible existence of a modulated phase. On the other hand, analyzing the properties of the full set of solutions of RGFE we obtain that the model could have at least eight nonuniversal ordered phases in the small Γ 0 limit, four of these being modulated and the remaining four nonmodulated.

Derivations in covariant representations of C*-algebras

Abstract: Soit α une action d'un groupe G localement compact abelien ou compact sur une C * -algebre #7B-A et Π une representation de #7B-A qui induit une action α÷o de Π(#7B-A) a partir de α. Si δ est une * -derivation localement bornee dans #7B-A definie sur #7B-A F , alors il existe une * -derivation localement σ-faiblement continue δ~ dans Π(#7B-A) definie sur Π(#7B-A) F telle que δoΠ⊃Πoδ