Showing papers on "Operator algebra published in 2011"

Generalized Vertex Algebras and Relative Vertex Operators

Abstract: 1. Introduction. 2. The setting. 3. Relative untwisted vertex operators. 4. Quotient vertex operators. 5. A Jacobi identity for relative untwisted vertex operators. 6. Generalized vertex operator algebras and their modules. 7. Duality for generalized vertex operator algebras. 8. Monodromy representations of braid groups. 9. Generalized vertex algebras and duality. 10. Tensor products. 11. Intertwining operators. 12. Abelian intertwining algebras, third cohomology and duality. 13. Affine Lie algebras and vertex operator algebras. 14. Z-algebras and parafermion algebras. References. List of frequently-used symbols, in order of appearance.

A Polynomial Approach to Linear Algebra

Abstract: Preliminaries.- Linear Spaces.- Determinants.- Linear Transformations.- The Shift Operator.- Structure Theory of Linear Transformations.- Inner Product Spaces.- Quadratic Forms.- Stability.- Elements of System Theory.- Hankel Norm Approximation.

Complete set of cut-and-join operators in the Hurwitz-Kontsevich theory

Abstract: We define cut-and-join operators in Hurwitz theory for merging two branch points of an arbitrary type. These operators have two alternative descriptions: (1) the GL characters are their eigenfunctions and the symmetric group characters are their eigenvalues; (2) they can be represented as W-type differential operators (in particular, acting on the time variables in the Hurwitz-Kontsevich τ-function). The operators have the simplest form when expressed in terms of the Miwa variables. They form an important commutative associative algebra, a universal Hurwitz algebra, generalizing all group algebra centers of particular symmetric groups used to describe the universal Hurwitz numbers of particular orders. This algebra expresses arbitrary Hurwitz numbers as values of a distinguished linear form on the linear space of Young diagrams evaluated on the product of all diagrams characterizing particular ramification points of the branched covering.

Deep beauty : understanding the quantum world through mathematical innovation

Abstract: Part I. Beyond the Hilbert Space Formalism: Category Theory: 1. A prehistory of n-categorical physics John C. Baez and Aaron Lauda 2. A universe of processes and some of its guises Bob Coecke 3. Topos methods in the foundations of physics Chris J. Isham 4. The physical interpretation of daseinisation Andreas Doring 5. Classical and quantum observables Hans F. de Groote 6. Bohrification Chris Heunen, Nicolaas P. Landsman and Bas Spitters Part II. Beyond the Hilbert Space Formalism: Operator Algebras: 7. Yet more ado about nothing: the remarkable relativistic vacuum state Stephen J. Summers 8. Einstein meets von Neumann: locality and operational independence in algebraic quantum field theory Miklos Redei Part III. Behind the Hilbert Space Formalism: 9. Quantum theory and beyond: is entanglement special? Borivoje Dakic and Caslav Brukner 10. Is Von Neumann's 'no hidden variables' proof silly? Jeffrey Bub 11. Foliable operational structures for general probabilistic theories Lucien Hardy 12. The strong free will theorem John H. Conway and Simon Kochen.

Representation Theory of the Virasoro Algebra

Abstract: Preliminary.- Classification of Harish-Chandra Modules.- The Jantzen Filtration.- Determinant Formulae.- Verma Modules I: Preliminaries.- Verma Modules II: Structure Theorem.- A Duality among Verma Modules.- Fock Modules.- Rational Vertex Operator Algebras.- Coset Constructions for sl2.- Unitarisable Harish-Chandra Modules.- Homological Algebras.- Lie p-algebras.- Vertex Operator Algebras.

The bond-algebraic approach to dualities

Abstract: An algebraic theory of dualities is developed based on the notion of bond algebras. It deals with classical and quantum dualities in a unified fashion explaining the precise connection between quantum dualities and the low temperature (strong-coupling)/high temperature (weak-coupling) dualities of classical statistical mechanics (or (Euclidean) path integrals). Its range of applications includes discrete lattice, continuum field and gauge theories. Dualities are revealed to be local, structure-preserving mappings between model-specific bond algebras that can be implemented as unitary transformations, or partial isometries if gauge symmetries are involved. This characterization permits us to search systematically for dualities and self-dualities in quantum models of arbitrary system size, dimensionality and complexity, and any classical model admitting a transfer matrix or operator representation. In particular, special dualities such as exact dimensional reduction, emergent and gauge-reducing dualities th...

Operator Algebras for Multivariable Dynamics

Abstract: Let $X$ be a locally compact Hausdorff space with $n$ proper continuous self maps $\sigma_i:X \to X$ for $1 \le i \le n$. To this the authors associate two conjugacy operator algebras which emerge as the natural candidates for the universal algebra of the system, the tensor algebra $\mathcal{A}(X,\tau)$ and the semicrossed product $\mathrm{C}_0(X)\times_\tau\mathbb{F}_n^+$. They develop the necessary dilation theory for both models. In particular, they exhibit an explicit family of boundary representations which determine the C*-envelope of the tensor algebra.|Let $X$ be a locally compact Hausdorff space with $n$ proper continuous self maps $\sigma_i:X \to X$ for $1 \le i \le n$. To this the authors associate two conjugacy operator algebras which emerge as the natural candidates for the universal algebra of the system, the tensor algebra $\mathcal{A}(X,\tau)$ and the semicrossed product $\mathrm{C}_0(X)\times_\tau\mathbb{F}_n^+$. They develop the necessary dilation theory for both models. In particular, they exhibit an explicit family of boundary representations which determine the C*-envelope of the tensor algebra.

Model theory of operator algebras III: Elementary equivalence and II_1 factors

Abstract: We use continuous model theory to obtain several results concerning isomorphisms and embeddings between II_1 factors and their ultrapowers. Among other things, we show that for any II_1 factor M, there are continuum many nonisomorphic separable II_1 factors that have an ultrapower isomorphic to an ultrapower of M. We also give a poor man's resolution of the Connes Embedding Problem: there exists a separable II_1 factor such that all II_1 factors embed into one of its ultrapowers.

Field theories with defects and the centre functor

Abstract: This note is intended as an introduction to the functorial formulation of quantum field theories with defects. After some remarks about models in general dimension, we restrict ourselves to two dimensions – the lowest dimension in which interesting field theories with defects exist. We study in some detail the simplest example of such a model, namely a topological field theory with defects which we describe via lattice TFT. Finally, we give an application in algebra, where the defect TFT provides us with a functorial definition of the centre of an algebra. This involves changing the target category of commutative algebras into a bicategory. Throughout this paper, we emphasise the role of higher categories – in our case bicategories – in the description of field theories with defects.

Partial actions: a survey

Abstract: We give a short survey on partial actions, partial representations and related notions. In the theory of C ∗ -algebras a concept of a partial action was introduced as an efficient tool of their study, permitting to characterize various important classes of C ∗ -algebras as crossed products by partial actions. Such a characterization made it possible to obtain relevant results on K-theory, ideal structure and representa- tions of the algebras under consideration. Also amenability questions, especially amenability of C ∗ -algebraic bundles (also called Fell bundles) 1 , were successfully investigated using partial actions and the related concept of a partial representa- tion. Crossed products play a central role in the rich interaction between operator algebras and dynamical systems, and partial actions on C ∗ -algebras (= partial C ∗ - dynamical systems) provide an appropriate approach to C ∗ -algebras generated by partial isometries. Amongst prominent classes of C ∗ -algebras endowed with the structure of non-trivial crossed products by partial actions one may list the Bunce- Deddens and the Bunce-Deddens-Toeplitz algebras (42), the approximately finite dimensional algebras (43), the Toeplitz algebras of quasi-ordered groups, as well as the Cuntz-Krieger algebras (52), (88). The notion of a partial action on a C ∗ -algebra appeared first time in the li- terature in Ruy Exel's paper (41 )i n whichC ∗ -algebraic crossed products by a partial automorphism (equivalently, by a partial action of the infinite cyclic group) were introduced and studied from the point of view of their internal structure, K- theory and representations. Motivated by a dynamical system point of view, Exel's main purpose was to develop a method which allows to describe the structure of C ∗ - algebras possessing actions of the circle group. The possibility for a straightforward generalization of the main construction was also noted. Exel's paper was followed by K. McClanahan's article (74) in which the formal definition of a C ∗ -crossed pro- duct by a partial action of a discrete group was given, permitting to obtain further K-theoretic results. The general notion of a (continuous) twisted partial action of

Operator system quotients of matrix algebras and their tensor products

Abstract: An operator system modulo the kernel of a completely positive linear map of the operator system gives rise to an operator system quotient. In this paper, operator system quotients and quotient maps of certain matrix algebras are considered. Some applications to operator algebra theory are given, including a new proof of Kirchberg's theorem on the tensor product of B(H) with the group C*-algebra of a countable free group. We also show that an affirmative solution to the Connes Embedding Problem is implied by various matrix-theoretic problems, and we give a new characterisation of unital C*-algebras that have the weak expectation property.

Why be normal

Abstract: Suppose we have an algebra A of quantum observables. What virtues must a function ω:A→C exhibit in order to qualify as a quantum state? One virtue familiar from density operator states is countable additivity: a density operator ρ on H determines a countably additive probability distribution over H's closed subspaces; such a probability distribution corresponds to what's known as a normal state on the von Neumann algebra B(H) of bounded operators on the Hilbert space. This essay investigates the virtue of normality, by presenting a number of physical situations and/or interpretive impulses that might tempt one to acknowledge states that are not normal, and by adducing reasons to resist these temptations.

Quantum Geometry on Quantum Spacetime: Distance, Area and Volume Operators

Abstract: We develop the first steps towards an analysis of geometry on the quantum spacetime proposed in Doplicher et al. (Commun Math Phys 172:187-220, 1995). The homogeneous elements of the universal differential algebra are naturally identified with operators living in tensor powers of Quantum Spacetime; this allows us to compute their spectra. In particular, we consider operators that can be interpreted as distances, areas, 3- and 4-volumes. The Minkowski distance operator between two independent events is shown to have pure Lebesgue spectrum with infinite multiplicity. The Euclidean distance operator is shown to have spectrum bounded below by a constant of the order of the Planck length. The corresponding statement is proved also for both the space-space and space-time area operators, as well as for the Euclidean length of the vector representing the 3-volume operators. However, the space 3-volume operator (the time component of that vector) is shown to have spectrum equal to the whole complex plane. All these operators are normal, while the distance operators are also selfadjoint. The Lorentz invariant spacetime volume operator, representing the 4-volume spanned by five independent events, is shown to be normal. Its spectrum is pure point with a finite distance (of the order of the fourth power of the Planck length) away from the origin. The mathematical formalism apt to these problems is developed and its relation to a general formulation of Gauge Theories on Quantum Spaces is outlined. As a byprod- uct, a Hodge Duality between the absolute differential and the Hochschild boundary is pointed out.

Noncommutative spectral geometry, algebra doubling and the seeds of quantization

Abstract: is presented. It is shown that the doubling of the algebra is related to dissipation and to the gauge structure of the theory, the gauge eld acting as a reservoir for the matter eld. In a regime of completely deterministic dynamics, dissipation appears to play a key r^ole in the quantization of the theory, according to ’t Hooft’s conjecture. It is thus argued that the noncommutative spectral geometry classical construction carries implicit in its feature of the doubling of the algebra the seeds of quantization.

Quasibosons composed of two q-fermions: realization by deformed oscillators

Abstract: Composite bosons, here called quasibosons (e.g. mesons, excitons, etc), occur in various physical situations. Quasibosons differ from bosons or fermions as their creation and annihilation operators obey non-standard commutation relations, even for the ?fermion+fermion? composites. Our aim is to realize the operator algebra of quasibosons composed of two fermions or two q-fermions (q-deformed fermions) by the respective operators of deformed oscillators, the widely studied objects. For this, the restrictions on quasiboson creation/annihilation operators and on the deformed oscillator (deformed boson) algebra are obtained. Their resolving proves the uniqueness of the family of deformations and gives explicitly the deformation structure function (DSF) which provides the desired realization. In the case of two fermions as constituents, such realization is achieved when the DSF is a quadratic polynomial in the number operator. In the case of two q-fermions, q ? 1, the obtained DSF inherits the parameter q and does not continuously converge when q ? 1 to the DSF of the first case.

Dilation Theory, Commutant Lifting and Semicrossed Products

Abstract: We take a new look at dilation theory for nonself-adjoint operator algebras Among the extremal (co)extensions of a repre- sentation, there is a special property of being fully extremal This allows a refinement of some of the classical notions which are im- portant when one moves away from standard examples We show that many algebras including graph algebras and tensor algebras of C*-correspondences have the semi-Dirichlet property which collapses these notions and explains why they have a better dilation theory This leads to variations of the notions of commutant lifting and Ando's theorem This is applied to the study of semicrossed products by au- tomorphisms, and endomorphisms which lift to the C*-envelope In particular, we obtain several general theorems which allow one to conclude that semicrossed products of an operator algebra naturally imbed completely isometrically into the semicrossed product of its C*- envelope, and the C*-envelopes of these two algebras are the same

Determination of the Schmidt number

Abstract: Optimized, necessary, and sufficient conditions for the identification of the Schmidt number will be derived in terms of general Hermitian operators. These conditions apply to arbitrary mixed quantum states. The optimization procedure delivers equations similar to the eigenvalue problem of an operator. The properties of the solution of these equations will be studied. We solve these equations for classes of operators. The solutions will be applied to phase randomized two-mode squeezed-vacuum states in continuous variable systems.

Topological Graph Polynomial and Quantum Field Theory Part II: Mehler Kernel Theories

Abstract: We define a new topological polynomial extending the Bollobas–Riordan one, which obeys a four-term reduction relation of the deletion/contraction type and has a natural behaviour under partial duality. This allows to write down a completely explicit combinatorial evaluation of the polynomials, occurring in the parametric representation of the non-commutative Grosse–Wulkenhaar quantum field theory. An explicit solution of the parametric representation for commutative field theories based on the Mehler kernel is also provided.

K-homology class of the dirac operator on a compact quantum group

Abstract: By a result of Nagy, the C � -algebra of continuous func- tions on the q-deformation Gq of a simply connected semisimple com- pact Lie group G is KK-equivalent to C(G). We show that under this equivalence the K-homology class of the Dirac operator on Gq, which we constructed in an earlier paper, corresponds to that of the classi- cal Dirac operator. Along the way we prove that for an appropriate choice of isomorphisms between completions of Uqg and Ug a family of Drinfeld twists relating the deformed and classical coproducts can be chosen to be continuous in q.

Star product realizations of kappa-Minkowski space

Abstract: We define a family of star products and involutions associated with $\kappa$-Minkowski space. Applying corresponding quantization maps we show that these star products restricted to a certain space of Schwartz functions have isomorphic Banach algebra completions. For two particular star products it is demonstrated that they can be extended to a class of polynomially bounded smooth functions allowing a realization of the full Hopf algebra structure on $\kappa$-Minkowski space. Furthermore, we give an explicit realization of the action of the $\kappa$-Poincare algebra as an involutive Hopf algebra on this representation of $\kappa$-Minkowski space and initiate a study of its properties.

Quasibosons composed of two q-fermions: realization by deformed oscillators

Abstract: Composite bosons, here called {\it quasibosons} (e.g. mesons, excitons, etc.), occur in various physical situations. Quasibosons differ from bosons or fermions as their creation and annihilation operators obey non-standard commutation relations, even for the "fermion+fermion" composites. Our aim is to realize the operator algebra of quasibosons composed of two fermions or two q-fermions (q-deformed fermions) by the respective operators of deformed oscillators, the widely studied objects. For this, the restrictions on quasiboson creation/annihilation operators and on the deformed oscillator (deformed boson) algebra are obtained. Their resolving proves uniqueness of the family of deformations and gives explicitly the deformation structure function (DSF) which provides the desired realization. In case of two fermions as constituents, such realization is achieved when the DSF is quadratic polynomial in the number operator. In the case of two q-fermions, q eq 1, the obtained DSF inherits the parameter q and does not continuously converge when q\to 1 to the DSF of the first case.

Wandering vectors and the reflexivity of free semigroup algebras

Abstract: A free semigroup algebra S is the weak-operator-closed (non-self-adjoint) operator algebra generated by n isometries with pairwise orthogonal ranges. A unit vector x is said to be wandering for S if the set of images of x under non-commuting words in the generators of S is orthonormal. We establish the following dichotomy: either a free semigroup algebra has a wandering vector, or it is a von Neumann algebra. Consequences include that every free semigroup algebra is reflexive, and that certain free semigroup algebras are hyper-reflexive with a very small hyper-reflexivity constant.

Generalized Effect Algebras of Positive Operators Densely Defined on Hilbert Spaces

Abstract: Axioms of quantum structures, motivated by properties of some sets of linear operators in Hilbert spaces are studied. Namely, we consider examples of sets of positive linear operators defined on a dense linear subspace D in a (complex) Hilbert space ℋ. Some of these operators may have a physical meaning in quantum mechanics. We prove that the set of all positive linear operators with fixed such D and ℋ form a generalized effect algebra with respect to the usual addition of operators. Some sub-algebras are also mentioned. Moreover, on a set of all positive linear operators densely defined in an infinite dimensional complex Hilbert space, the partial binary operation is defined making this set a generalized effect algebra.