Showing papers on "Operator algebra published in 1988"
TL;DR: In this article, the equality between the restriction of the Adler-Manin-Wodzicki residue or non-commutative residue to pseudodifferential operators of order −n on ann-dimensional compact manifoldM, with the trace which J. Dixmier constructed on the Macaev ideal was established.
Abstract: We establish the equality between the restriction of the Adler-Manin-Wodzicki residue or non-commutative residue to pseudodifferential operators of order −n on ann-dimensional compact manifoldM, with the trace which J. Dixmier constructed on the Macaev ideal. We then use the latter trace to recover the Yang Mills interaction in the context of non-commutative differential geometry.
343 citations
TL;DR: In this article, a two-parameter algebra-co-algebra KAB is proposed to reformulate the classical and quantum mechanics of a particle moving on a homogeneous spacetime.
Abstract: Applies ideas of non-commutative geometry to reformulate the classical and quantum mechanics of a particle moving on a homogeneous spacetime. The reformulation maintains an interesting symmetry between observables and states in the form of a Hopf algebra structure. In the simplest example both the dynamics and quantum mechanics are completely determined by the Hopf algebra consideration. The simplest example is a two-parameter algebra-co-algebra KAB which is the unique Hopf algebra extension, such as is possible, of the self-dual Hopf algebra of functions on flat phase space C(R*R). In the limit (A=0,B) the author recovers functions on a curved phase space with curvature proportional to B2 and in another limit (A= infinity , B= infinity ), A/B=h(cross), the author recovers quantum mechanics on R>or=0 with an absorbing wall at the origin. The algebra in this way corresponds to a toy model of quantum mechanics of a particle in one space dimension combined with gravity-like forces. It has an interesting Z2 symmetry interchanging A to or from B, and thereby, in some sense, the quantum element with the geometric element. The compatibility conditions that are solved are a generalisation of the classical Yang-Baxter equations.
268 citations
TL;DR: In this article, a class of 2D statistical mechanics models known as IRF models can be viewed as a subalgebra of the operator algebra of vertex models, and an explicit intertwiner between two representations of this sub-algebra is obtained.
Abstract: We show that a class of 2D statistical mechanics models known as IRF models can be viewed as a subalgebra of the operator algebra of vertex models. Extending the Wigner calculus to quantum groups, we obtain an explicit intertwiner between two representations of this subalgebra.
208 citations
TL;DR: In this paper, the Moyal *-algebra is defined as an operator algebra over the space of smooth declining functions either on the configuration space or on the phase space itself.
Abstract: The topology of the Moyal *‐algebra may be defined in three ways: the algebra may be regarded as an operator algebra over the space of smooth declining functions either on the configuration space or on the phase space itself; or one may construct the *‐algebra via a filtration of Hilbert spaces (or other Banach spaces) of distributions. The equivalence of the three topologies thereby obtained is proved. As a consequence, by filtrating the space of tempered distributions by Banach subspaces, new sufficient conditions are given for a phase‐space function to correspond to a trace‐class operator via the Weyl correspondence rule.
129 citations
27 Oct 1988
TL;DR: In this paper, a spatial operator algebra for modeling, control and trajectory design of manipulators is discussed, which is based on linear operators whose domain and range spaces consist of forces, moments, velocities, and accelerations.
Abstract: A recently developed spatial operator algebra for modeling, control and trajectory design of manipulators is discussed. The elements of this algebra are linear operators whose domain and range spaces consist of forces, moments, velocities, and accelerations. The operators themselves are elements in the algebra of linear bounded operators. The effect of these operators when operating on elements in the domain is equivalent to a spatial recursion along the span of a manipulator. Inversion of operators can be efficiently obtained via techniques of spatially recursive filtering and smoothing. The operator algebra provides a high-level framework for describing the dynamic and kinematic behavior of a manipulator and for the corresponding control and trajectory design algorithms. Expressions interpreted within the algebraic framework lead to enhanced conceptual and physical understanding of manipulator dynamics and kinematics. Furthermore, implementable recursive algorithms can be immediately derived from the abstract operator expressions by inspection. Thus, the transition from an abstract problem formulation and solution to the detailed mechanization of specific algorithms is greatly simplified. This paper discusses the analytical formulation of the operator algebra, as well as its implementation in the Ada programming language.
89 citations
01 Jan 1988
TL;DR: The relationship between the conformal field theories and the soliton equations (KdV, MKdV and Sine-Gordon, etc.) at both quantum and classical levels is discussed in this article.
Abstract: The relationship between the conformal field theories and the soliton equations (KdV, MKdV and Sine–Gordon, etc.) at both quantum and classical levels is discussed. The quantum Sine–Gordon theory is formulated canonically. Its Hamiltonian is the vertex operator with respect to the Feigin–Fuchs–Miura form of the Virasoro algebra with central charge $c\le1$. It is found that the quantum conserved quantities of the Sine–Gordon-MKdV hierarchy are expressed as polynomial functions of the Virasoro generators. In other words, an infinite set of mutually commutative polynomial functions of the Virasoro generators is obtained. A very simple recursion formula for the quantum conserved quantities is found for the special case of $\beta^2_c=8\pi$ ($\beta_c$ is the coupling constant in Coleman’s theory of quantum Sine–Gordon).
85 citations
TL;DR: In this paper, it was shown that the only ergodic actions of SU(2) are on Type I von Neumann algebras, using the theory developed in [8].
Abstract: We prove that the only ergodic actions ofSU(2) are on Type I von Neumann algebras, using the theory developed in [8].
82 citations
57 citations
01 Jul 1988
TL;DR: In this paper, the authors investigate geometrical properties of algebra norms on ⊗ ℬ and show that algebra norm α(u.v) ≤ ≤ α (u).α(v).
Abstract: When and ℬ are C*-algebras their algebraic tensor product ⊗ ℬ is a *-algebra in a natural way. Until recently, work on tensor products of C*-algebras has concentrated on norms α which make the completion ⊗α ℬ into a C*-algebra. The crucial role played by the Haagerup norm in the theory of operator spaces and completely bounded maps has produced some interest in more general norms (see [8; 12]). In this paper we investigate geometrical properties of algebra norms on ⊗ ℬ. By an ‘algebra norm’ we mean a norm which is sub-multiplicative: α(u.v) ≤ ≤ α(u).α(v).
53 citations
TL;DR: In this article, theoreme de structure for des algebres sous diagonales contenant une sous algebre de Cartan was demontre.
Abstract: On demontre un theoreme de structure pour des algebres sous diagonales contenant une sous algebre de Cartan. On etudie les isomorphismes pour ces algebres
50 citations
TL;DR: In this article, it was shown that two continuous maps are conjugate if and only if some conjugacy algebra of one is isomorphic to some conjjugacy for the other.
TL;DR: In this paper, the maximal ideal space of Banach algebras generated by two idempotents and by a certain flip operator was determined and the corresponding symbol was given.
Abstract: It is proved that in Banach algebras generated by two idempotents and, perhaps, by a certain flip operator the standard identity F4 is fulfilled. The maximal ideal space of such algebras is determined and the corresponding symbol is given. By means of local techniques these results are applied to obtain a symbol calculus for singular integral operators with Carleman shift (changing the orientation) in weighted Banach spaces.
TL;DR: An algebraic definition of the basic quantum process for the noncommutative stochastic calculus is given in terms of the Fock representation of a Lie ⋆-algebra of matrices in a pseudo-Euclidean space as discussed by the authors.
Abstract: An algebraic definition of the basic quantum process for the noncommutative stochastic calculus is given in terms of the Fock representation of a Lie ⋆-algebra of matrices in a pseudo-Euclidean space. An operator definition of the quantum stochastic integral is given and its continuity is proved in a projective limit uniform operator topology. A new form of quantum stochastic equations, revealing the ⋆-algebraic structure of quantum Ito's formula, is given.
TL;DR: In this article, a number of equivalent conditions which characterize the trace among the linear functionals on the matrix algebra are presented, and some of these results are extended to more general operator algebras.
TL;DR: In this paper, the two-dimensional Hall effect with a random potential was studied and the Hall conductivity was identified as a geometric invariant associated with an algebra of observables.
Abstract: We study the two-dimensional Hall effect with a random potential. The Hall conductivity is identified as a geometric invariant associated with an algebra of observables. Using the pairing betweenK-theory and cyclic cohomology theory, we identify this geometric invariant with a topological index, thereby giving the Hall conductivity a new interpretation.
TL;DR: Soit (X, φ) un systeme dynamique localement compact, and Z + Xφ Co(X) la sous-algebre close en norme du produit croise ZXφCo(X), engendre par les puissances non negatives de φ dans le cas φ homeomorphisme.
Abstract: Soit (X, φ) un systeme dynamique localement compact, et Z + Xφ Co(X) la sous-algebre close en norme du produit croise ZXφCo(X) engendre par les puissances non negatives de φ dans le cas φ homeomorphisme. On determine la structure ideale de ces algebres dans le cas ou φ agit librement
TL;DR: In this article, the Dixmier-Douady invariant is defined for continuous trace C*-algebras with spectrum X. The invariant can be seen as a special case of strong Morita equivalence.
Abstract: Let X be a locally compact Hausdorff space. The graded Morita equivalence classes of separable, Z2-graded, continuous trace C*-algebras which have spectrum X form a group, GBr??(X), the infinite-dimensional graded Brauer group of X. Techniques from algebraic topology are used to prove that GBr??(X) is isomorphic via an isomorphism w to the direct sum t1 (X; Z2) E 13 (X; Z). The group GBr? (X) includes as a subgroup the ungraded continuous trace C*-algebras, and the Dixmier-Douady invariant of such an ungraded C*-algebra is its image in H3 (X; Z) under w. Introduction. The study of graded C*-algebras has become particularly important since G. G. Kasparov's development of KK-theory for operator algebras [18]. In this paper, separable, Z2-graded, continuous trace C*-algebras are classified. The graded Morita equivalence classes of such algebras whose spectra are all the same locally compact Hausdorff space X form a group, called the infinitedimensional graded Brauer group of X and denoted by GBr??(X). Two invariants defined on GBr??(X) provide useful insights into the structure of these C*-algebras and relate the results presented here to previous work. The constructions of J. Dixmier and A. Douady [3, 4, 51 form an important framework for the graded classification. Let X be a locally compact Hausdorff space, with countable base. Dixmier and Douady considered separable, stable, continuous trace C*-algebras, with spectrum X. There is a canonical way to associate such an algebra A with a fiber bundle (A over X with fiber X(X), the compact operators on an infinite-dimensional separable Hilbert space. Let 7 W(R') be the projective unitary group of , and let fI* (X; G) denote the tech cohomology of X with coefficients in the sheaf of germs of continuous functions from X to G, for G a group. Then the isomorphism class of (A is an element of fI (X; 37 &(X)), which can be shown to be isomorphic to f3I(X; Z). They defined the Dixmier-Douady invariant 6(A) E H13(X; Z) of the algebra A, and proved that the invariant defines a one-to-one correspondence between isomorphism classes of such algebras and the elements of H3 (X; Z). Consider now the collection of graded, separable, continuous trace C*-algebras, all with spectrum X. We will define GBr??(X) as the set of equivalence classes of all such C*-algebras under graded Morita equivalence, which is the graded version of strong Morita equivalence defined by M. Rieffel [22, 23]. It is important to note, Received by the editors December 15, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 46M20, 46L80, 46L35; Secondary 55R10.
01 May 1988
TL;DR: In this article, the structure constants of a two-dimensional superconformal field theory closed algebra under operator product expansion are calculated. But they do not consider the case where the fields are generated by degenerate fields.
Abstract: Degenerate fields in two-dimensional superconformal field theory form a closed algebra under operator product expansion. All structure constants of this operator algebra are calculated.
01 Mar 1988
TL;DR: In this paper, a groupe localement compact is defined as "a group of operateurs lineaires that satisfy a propriete invariant de dimension finie T(n) for des sous-espaces a n dimensions contenus dans un sousensemble X d'un espace separe localement convexe E quand L 1 (G) est represente comme des operateurs LINE this paper.
Abstract: Soit G un groupe localement compact. On demontre que G est moyennable, si et seulement si l'algebre de groupe L 1 (G) satisfait une propriete de sous-espace invariant de dimension finie T(n) pour des sous-espaces a n dimensions contenus dans un sous-ensemble X d'un espace separe localement convexe E quand L 1 (G) est represente comme des operateurs lineaires continus sur E
TL;DR: In this paper, a truncated version of the constraint algebra for the p -brane is considered and conditions on the dimension of the embedding space are derived for the membrane.
01 Jan 1988
TL;DR: In this paper, a general covariant quasilocal algebra associated with the massive free field is presented, in which maximal ideals are viewed as algebraic representatives of dynamical equations or Lagrangians.
Abstract: We give an example of a generally covariant quasilocal algebra associated with the massive free field. Maximal, two-sided ideals of this algebra are algebraic representatives of external metric fields. In some sense, this algebra may be regarded as a concrete realization of Ekstein's ideas of presymmetry in quantum field theory. Using ideas from our example and from usual algebraic quantum field theory, we discuss a generalized scheme, in which maximal ideals are viewed as algebraic representatives of dynamical equations or Lagrangians. The considered frame is no quantum gravity, but may lead to further insight into the relation between quantum theory and space-time geometry.
01 Apr 1988
Abstract: In [2] the authors described all weakly closed _W-submodules of L(H) for a nest algebra v in terms of order homomorphisms of Lat-'. In this paper we prove that for any reflexive algebra v which is a-weakly generated by rank-one operators in X, every a-weakly closed s/-submodule can be characterized by an order homomorphism of Lat-W. In the case when v is a reflexive algebra with a completely distributive subspace lattice and / is a a-weakly closed ideal of X, we obtain necessary and sufficient conditions for the commutant of v modulo X to be equal to AlgLatf. Let H be a complex Hilbert space, and let L(H) be the set of all bounded linear operators on H. The terminology and notation of this paper concerning nest algebras and reflexive subspaces of L(H) may be found in [3]. Let v be a reflexive subalgebra of L(H). Suppose that E is an order homomorphism of Lat _W into itself (i.e. E < F implies E < F), where Lat &' is the set of all invariant projections for -W. Then the set //'X = {T c L(H): (I-E)TE = 0 for all E E Lat -W} is clearly a weakly closed -W -subrnodule of L(H). J. A. Erdos and S. C. Power in [2] proved that any weakly closed -W-submodule of L(H) for a nest algebra v is of the above form. Here we prove that this is also true for any reflexive algebra v which is a-weakly generated by rank-one operators in X. The following result is due to J. Kraus and D. R. Larson [3]. THEOREM 1. Let v be a unital a-weakly closed algebra which is a-weakly generated by rank-one operators in -W. Then every a-weakly closed left or right module of is reflexive. THEOREM 2. Let v be as in the above theorem, and let X be a a-weakly closed v -submodule of L (H). Then X' has the form X = {T c L(H): (I E)TE = 0 for all E c Lat-'}, where E f-* E is some order homomorphism of Lat-W into itself. PROOF. For any E c Lat-W, let E be the orthogonal projection onto [/'EH] = V{ran(XE): X c X}. Since I is an -'-submodule, E is invariant under -v and clearly E ~-+ E is an order homomorphism. Let -I = {T c L(H): (I E)TE = 0 for all E c Lat-&}. It is obvious that X D X'. Conversely, if T c IV, then (I E)TE = 0, so [TEH] C [EH] = [vfEH] for any E E Lat-'. Now for any Received by the editors March 20, 1987 and, in revised form, December 7, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 47D25, 47D15; Secondary 47B47.
TL;DR: In this paper, the structure of the Hilbert space for the quantized coupled Dirac-Yang-Mills system is discussed and the existence of the vacuum vector and the cancellation of commutator anomalies is described in terms of complex line bundles over infinite-dimensional Grassmannians.
Abstract: The structure of the current algebra representation in the state space of fermions in an external Yang-Mills field in 3+1 space-time dimensions is analyzed; the topology of the vector space is determined by a countable family of semi-definite inner products. We show that there is no hermitian non-trivial Hilbert space representation such that the energy is bounded from below. The structure of the Hilbert space for the quantized coupled Dirac-Yang-Mills system is discussed and the existence of the vacuum vector and the cancellation of commutator anomalies is described in terms of complex line bundles over infinite-dimensional Grassmannians.
TL;DR: The nature of diagrammatic perturbation theory in relativistic field theory at nonzero temperature is investigated and by operator-algebraic techniques it is found that the conventional method is inconsistent, and an essentially unique alternative is given.
Abstract: By use of general principles (Kubo-Martin-Schwinger condition, breakdown of Lorentz invariance, triviality arguments), the nature of diagrammatic perturbation theory in relativistic field theory at nonzero temperature is investigated. By operator-algebraic techniques it is found that the conventional method is inconsistent, and an essentially unique alternative is given.
24 Aug 1988
TL;DR: In this paper, a spatial operator algebra for modeling, control, and trajectory design of manipulators is discussed, which is a high-level framework for describing the dynamic and kinematic behavior of a manipulator.
Abstract: A spatial operator algebra for modeling, control, and trajectory design of manipulators is discussed. The elements of this algebra are linear operators whose domain and range spaces consist of forces, moments, velocities, and accelerations. The operators themselves are elements in the algebra of linear bounded operators. The effect of these operators when operating on elements in the domain is equivalent to a spatial recursion along the span of a manipulator. Inversion of operators can be efficiently obtained via techniques of spatially recursive filtering and smoothing. The operator algebra provides a high-level framework for describing the dynamic and kinematic behavior of a manipulator and for developing corresponding control and trajectory design algorithms. Expressions interpreted within the operator algorithm framework led to enhanced conceptual and physical understanding of manipulator dynamics and kinematics. Furthermore, implementable recursive algorithms can be immediately derived from the high-level operator expressions by inspection. Thus, the transition from an abstract problem formulation and solution to the detailed mechanization of specific algorithms has been greatly simplified. The analytical formulation of the operator algebra and its implementation in Ada are discussed. >
Dissertation•
01 Jan 1988
TL;DR: In this paper, a generalization of the Frenkel-Kac-Segal mechanism to higher dimension operators is presented, which is a consequence of the duality of dimension one operators of an analytic string compactified on a certain torus.
Abstract: This work is principally concerned with the operator approach to the orbifold compactification of the bosonic string. Of particular importance to operator formalism is the con formal structure and the operator product expansion. These are introduced and discussed in detail. The Frenkel-Kac-Segal mechanism is then examined and is shown to be a consequence of the duality of dimension one operators of an analytic bosonic string compactified on a certain torus. Possible generalizations to higher dimension operators are discussed, this includes the cross-bracket algebra which plays a central role in the vertex operator representation of Griess's algebra, and hence the Fischer-Griess Monster Group. The mechanism of compactification is then extended to orbifolds. The exposition includes a detailed account of the twisted sectors, especially of the zero-modes and the twisted operator cocycles. The conformal structure, vertex operators and correlation functions for twisted strings are then introduced. This leads to a discussion of the vertex operators which represent the emission of untwisted states. It is shown how these operators generate Kac-Moody algebras in the twisted sectors. The vertex operators which insert twisted states are then constructed, and their role as intertwining operators is explained. Of particular importance in this discussion is the role of the operator cocycles, which are seen to be crucial for the correct working of the twisted string emission vertices. The previously established formalism is then applied in detail to the reflection twist. This includes an explicit representation of the twisted operator cocycles by elements of an appropriate Clifford algebra and the elucidation of the operator algebra of the twisted emission vertices, for the ground and first excited states in the twisted sector. This motivates the 'enhancement mechanism', a generalization of the Frenkel-Kac-Segal mechanism, involving twisted string emission vertices, in dimensions 8, 16 and 24. associated with rank 8 Lie algebras, rank 16 Lie algebras and the cross-bracket algebra for the Leech lattice, respectively. Some of the relevant characters of the 'enhanced" modules are determined, and the connection of the cross-bracket algebra to the phenomenon of 'Monstrous Moonshine' and the Monster Group is explained. Algebra enhancement is then discussed from the greatly simplified shifted picture and extensions to higher order twists are considered. Finally, a comparison of this work with other recent research is given. In particular, the connection with the path integral formalism and the extension to general asymmetric orbifolds is discussed. The possibility of reformulating the moonshine module in a 'covaxiant' twenty-six dimensional setting is also considered.
TL;DR: The algebraic structure of parastatistics has been generalized and it is found to be consistent with supersymmetric quantum mechanics with supercharges constructed out of the generalized para-Bose and para-Fermi operators as mentioned in this paper.
Abstract: The algebraic structure of parastatistics has been generalized and it is found to be consistent with supersymmetric quantum mechanics with supercharges constructed out of the generalized para‐Bose and para‐Fermi operators. It is further shown that the operator algebra of generalized parastatistics offers a realization of the (graded) orthosymplectic group similar to that of orthogonal and symplectic groups using conventional parastatistics.