Showing papers on "Operator algebra published in 2007"

Crossed products of C*-algebras

Abstract: Locally compact groups Dynamical systems and crossed products Special cases and basic constructions Imprimitivity theorems Induced representations and induced ideals Orbits and quasi-orbits Properties of crossed products Ideal structure The proof of the Gootman-Rosenberg-Sauvageot theorem Amenable groups The Banach *-algebra $L^1(G,A)$ Bundles of $C*$-algebras Groups Representations of $C*$-algebras Direct integrals Effros's ideal center decomposition The Fell topology Miscellany Notation and Symbol Index Index Bibliography.

Nonrelativistic conformal field theories

Abstract: We study representations of the Schr\"odinger algebra in terms of operators in nonrelativistic conformal field theories. We prove a correspondence between primary operators and eigenstates of few-body systems in a harmonic potential. Using the correspondence we compute analytically the energy of fermions at unitarity in a harmonic potential near two and four spatial dimensions. We also compute the energy of anyons in a harmonic potential near the bosonic and fermionic limits.

Comments on operators with large spin

Abstract: We consider high spin operators. We give a general argument for the logarithmic scaling of their anomalous dimensions which is based on the symmetries of the problem. By an analytic continuation we can also see the origin of the double logarithmic divergence in the Sudakov factor. We show that the cusp anomalous dimension is the energy density for a flux configuration of the gauge theory on AdS3 × S1. We then focus on operators in = 4 super Yang Mills which carry large spin and SO(6) charge and show that in a particular limit their properties are described in terms of a bosonic O(6) sigma model. This can be used to make certain all loop computations in the string theory.

Introducing Cadabra: a symbolic computer algebra system for field theory problems

Abstract: Cadabra is a new computer algebra system designed specifically for the solution of problems encountered in field theory. It has extensive functionality for tensor polynomial simplification taking care of Bianchi and Schouten identities, for fermions and anti-commuting variables, Clifford algebras and Fierz transformations, implicit coordinate dependence, multiple index types and many other field theory related concepts. The input format is a subset of TeX and thus easy to learn. Both a command-line and a graphical interface are available. The present paper is an introduction to the program using several concrete problems from gravity, supergravity and quantum field theory.

Renormalized Quantum Yang-Mills Fields in Curved Spacetime

Abstract: We present a proof that quantum Yang-Mills theory can be consistently defined as a renormalized, perturbative quantum field theory on an arbitrary globally hyperbolic curved, Lorentzian spacetime. To this end, we construct the non-commutative algebra of observables, in the sense of formal power series, as well as a space of corresponding quantum states. The algebra contains all gauge invariant, renormalized, interacting quantum field operators (polynomials in the field strength and its derivatives), and all their relations such as commutation relations or operator product expansion. It can be viewed as a deformation quantization of the Poisson algebra of classical Yang-Mills theory equipped with the Peierls bracket. The algebra is constructed as the cohomology of an auxiliary algebra describing a gauge fixed theory with ghosts and anti-fields. A key technical difficulty is to establish a suitable hierarchy of Ward identities at the renormalized level that ensure conservation of the interacting BRST-current, and that the interacting BRST-charge is nilpotent. The algebra of physical interacting field observables is obtained as the cohomology of this charge. As a consequence of our constructions, we can prove that the operator product expansion closes on the space of gauge invariant operators. Similarly, the renormalization group flow is proved not to leave the space of gauge invariant operators.

Covariant realizations of kappa-deformed space

Abstract: We study a Lie algebra type κ-deformed space with an undeformed rotation algebra and commutative vector-like Dirac derivatives in a covariant way. The space deformation depends on an arbitrary vector. Infinitely many covariant realizations in terms of commuting coordinates of undeformed space and their derivatives are constructed. The corresponding coproducts and star products are found and related in a new way. All covariant realizations are physically equivalent. Specially, a few simple realizations are found and discussed. The scalar fields, invariants and the notion of invariant integration is discussed in the natural realization.

Ideal structure of C∗-algebras associated with C∗-correspondences

Abstract: We study the ideal structure of C*-algebras arising from C*-correspondences. We prove that gauge-invariant ideals of our C*-algebras are parameterized by certain pairs of ideals of original C*-algebras. We show that our C*-algebras have a nice property that should be possessed by a generalization of crossed products. Applications to crossed products by Hilbert C*-bimodules and relative Cuntz?Pimsner algebras are also discussed.

Topological strings in generalized complex space

Abstract: A two-dimensional topological sigma-model on a generalized Calabi-Yau target space $X$ is defined. The model is constructed in Batalin-Vilkovisky formalism using only a generalized complex structure $J$ and a pure spinor $\rho$ on $X$. In the present construction the algebra of $Q$-transformations automatically closes off-shell, the model transparently depends only on $J$, the algebra of observables and correlation functions for topologically trivial maps in genus zero are easily defined. The extended moduli space appears naturally. The familiar action of the twisted N=2 CFT can be recovered after a gauge fixing. In the open case, we consider an example of generalized deformation of complex structure by a holomorphic Poisson bivector $\beta$ and recover holomorphic noncommutative Kontsevich $*$-product.

Wedge-Local Quantum Fields and Noncommutative Minkowski Space

Abstract: Within the setting of a recently proposed model of quantum fields on noncommutative Minkowski space, the consequences of the consistent application of the proper, untwisted Poincare group as the symmetry group are investigated. The emergent model contains an infinite family of fields which are labelled by different noncommutativity parameters, and related to each other by Lorentz transformations. The relative localization properties of these fields are investigated, and it is shown that to each field one can assign a wedge-shaped localization region in Minkowski space. This assignment is consistent with the principles of covariance and locality, i.e. fields localized in spacelike separated wedges commute. Regarding the model as a non-local, but wedge-local, quantum field theory on ordinary (commutative) Minkowski spacetime, it is possible to determine two-particle S-matrix elements, which turn out to be non-trivial. Some partial negative results concerning the existence of observables with sharper localization properties are also obtained.

Pseudodifferential operators on manifolds with a Lie structure at infinity

Abstract: We define and study an algebra Ψ ∞,0,V (M0) of pseudodifferential operators canonically associated to a noncompact, Riemannian manifold M0 whose geometry at infinity is described by a Lie algebra of vector fields V on a compactification M of M0 to a compact manifold with corners. We show that the basic properties of the usual algebra of pseudodifferential operators on a compact manifold extend to Ψ ∞,0,V (M0). We also consider the algebra Diff ∗ (M0) of differential operators on M0 generated by V and C ∞ (M), and show that Ψ ∞,0,V (M0) is a microlocalization of Diff ∗ (M0). Our construction solves a problem posed by Melrose in 1990. Finally, we introduce and study semi-classical and “suspended” versions of the algebra Ψ ∞,0,V (M0).

Quantum brachistochrone problem and the geometry of the state space in pseudo-hermitian quantum mechanics

Abstract: A non-Hermitian operator with a real spectrum and a complete set of eigenvectors may serve as the Hamiltonian operator for a unitary quantum system provided that one makes an appropriate choice for the defining the inner product of physical Hilbert state. We study the consequences of such a choice for the representation of states in terms of projection operators and the geometry of the state space. This allows for a careful treatment of the quantum Brachistochrone problem and shows that it is indeed impossible to achieve faster unitary evolutions using PT-symmetric or other non-Hermitian Hamiltonians than those given by Hermitian Hamiltonians.

Selbstduale Vertexoperatorsuperalgebren und das Babymonster (Self-dual Vertex Operator Super Algebras and the Baby Monster)

Abstract: We investigate self-dual vertex operator algebras (VOAs) and super algebras (SVOAs). Using the genus one correlation functions, it is shown that self-dual SVOAs exist only for half-integral central charges. It is described how self-dual SVOAs can be constructed from self-dual VOAs of larger central charge. The analogy with integral lattices and binary codes is emphasized. One main result is the construction of the shorter Moonshine module, a self-dual SVOA of central charge 23.5 on which the Baby monster - the second largest sporadic simple group - acts by automorphisms. The shorter Moonshine module has the character q^(-47/48)*(1+ 4371q^(3/2)+ 96256q^2+ 1143745q^(5/2) +...) and is the "shorter cousin" of the Moonshine module. Its lattice and code analog are the shorter Leech lattice and shorter Golay code. We conjecture that the shorter Moonshine module is the unique SVOA with this character. The final chapter introduces the notion of extremal VOAs and SVOAs. These are self-dual (S)VOAs with character having the same first few coefficients as the vacuum representation of the Virasoro algebra of the same central charge. We show that extremal VOAs exist at least for the central charges 8, 16, 24, 32, 40 and that extremal SVOAs exist only for the central charges c=0.5, 1, ..., 7.5, 8, 12, 14, 15, 15.5, 23.5 and 24. Examples for c=24 (resp. 23.5) are the (shorter) Moonshine module. Again, our results are similar to results known for codes and lattices.

Full field algebras

Abstract: We introduce a notion of full field algebra which is essentially an algebraic formulation of the notion of genus-zero full conformal field theory. For any vertex operator algebras VL and VR, \({V^{L}\otimes V^{R}}\) is naturally a full field algebra and we introduce a notion of full field algebra over \({V^{L}\otimes V^{R}}\) . We study the structure of full field algebras over \({V^{L}\otimes V^{R}}\) using modules and intertwining operators for VL and VR. For a simple vertex operator algebra V satisfying certain natural finiteness and reductivity conditions needed for the Verlinde conjecture to hold, we construct a bilinear form on the space of intertwining operators for V and prove the nondegeneracy and other basic properties of this form. The proof of the nondegenracy of the bilinear form depends not only on the theory of intertwining operator algebras but also on the modular invariance for intertwining operator algebras through the use of the results obtained in the proof of the Verlinde conjecture by the first author. Using this nondegenerate bilinear form, we construct a full field algebra over \({V\otimes V}\) and an invariant bilinear form on this algebra.

Uniform algebra isomorphisms and peripheral multiplicativity

Abstract: Let φ: A→ B be a surjective operator between two uniform alge-(1) =1 we Show that if φ satisfies the peripheral multiplicativity bras with φ(1) = 1. We show that if if satisfies the peripheral multiplicativity condition σ π (φ(f)φ(g)) = σ π (fg) for all f,g ∈ A, where σ π (f) is the peripheral spectrum of f, then φis an isometric algebra isomorphism from A onto B. One of the consequences of this result is that any surjective, unital, and multiplicative operator that preserves the peripheral ranges of algebra elements is an isometric algebra isomorphism. We describe also the structure of general, not necessarily unital, surjective and peripherally multiplicative operators between uniform algebras.

Quantum energy inequalities and local covariance II: Categorical formulation

Abstract: We formulate quantum energy inequalities (QEIs) in the framework of locally covariant quantum field theory developed by Brunetti, Fredenhagen and Verch, which is based on notions taken from category theory. This leads to a new viewpoint on the QEIs, and also to the identification of a new structural property of locally covariant quantum field theory, which we call local physical equivalence. Covariant formulations of the numerical range and spectrum of locally covariant fields are given and investigated, and a new algebra of fields is identified, in which fields are treated independently of their realisation on particular spacetimes and manifestly covariant versions of the functional calculus may be formulated.

On the Structure of Framed Vertex Operator Algebras and Their Pointwise Frame Stabilizers

Abstract: In this paper, we study the structure of a general framed vertex operator algebra (VOA). We show that the structure codes (C,D) of a framed VOA V satisfy certain duality conditions. As a consequence, we prove that every framed VOA is a simple current extension of the associated binary code VOA V C . This result suggests the feasibility of classifying framed vertex operator algebras, at least if the central charge is small. In addition, the pointwise frame stabilizer of V is studied. We completely determine all automorphisms in the pointwise stabilizer, which are of order 1, 2 or 4. The 4A-twisted sector and the 4A-twisted orbifold theory of the famous moonshine VOA $$V^ atural$$ are also constructed explicitly. We verify that the top module of this twisted sector is of dimension 1 and of weight 3/4 and the VOA obtained by 4A-twisted orbifold construction of $$V^ atural$$ is isomorphic to $$V^ atural$$ itself.

Bivariant algebraic k-theory

Abstract: We show how methods from K-theory of operator algebras can be applied in a completely algebraic setting to define a bivariant, M1-stable, homotopy-invariant, excisive K- theory of algebras over a fixed unital ground ring H, (A, B) 7→ kk�(A, B), which is universal in the sense that it maps uniquely to any other such theory. It turns out kk is related to C. Weibel's homotopy algebraic K-theory, KH. We prove that, if H is commutative and A is central as an H-bimodule, then kk�(H, A) = KH�(A). We show further that some calculations from operator algebra KK-theory, such as the exact sequence of Pimsner-Voiculescu, carry over to algebraic kk.

GHZ States, Almost-Complex Structure and Yang---Baxter Equation

Abstract: Recent research suggests that there are natural connections between quantum information theory and the Yang---Baxter equation. In this paper, in terms of the almost-complex structure and with the help of its algebra, we define the Bell matrix to yield all the Greenberger---Horne---Zeilinger (GHZ) states from the product basis, prove it to form a unitary braid representation and presents a new type of solution of the quantum Yang---Baxter equation. We also study Yang---Baxterization, Hamiltonian, projectors, diagonalization, noncommutative geometry, quantum algebra and FRT dual algebra associated with this generalized Bell matrix.

Lie triple derivations on nest algebras

Abstract: Let δ be a Lie triple derivation from a nest algebra into an -bimodule ℳ. We show that if ℳ is a weak* closed operator algebra containing then there are an element S ∈ ℳ and a linear functional f on such that δ (A) = SA – AS + f (A)I for all A ∈ , and if ℳ is the ideal of all compact operators then there is a compact operator K such that δ (A) = KA – AK for all A ∈ . As applications, Lie derivations and Jordan derivations on nest algebras are characterized. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

On symmetry enhancement in the psu(1, 1|2) sector of N = 4 SYM

Abstract: Strong evidence indicates that the spectrum of planar anomalous dimensions of = 4 super Yang-Mills theory is given asymptotically by Bethe equations. A curious observation is that the Bethe equations for the (1, 1|2) subsector lead to very large degeneracies of 2M multiplets, which apparently do not follow from conventional integrable structures. In this article, we explain such degeneracies by constructing suitable conserved nonlocal generators acting on the spin chain. We propose that they generate a subalgebra of the loop algebra for the (2) automorphism of (1, 1|2). Then the degenerate multiplets of size 2M transform in irreducible tensor products of M two-dimensional evaluation representations of the loop algebra.

Operator-algebraic superridigity for SL n (ℤ), n ≥3

Abstract: For n≥3, let Γ=SL n (ℤ). We prove the following superridigity result for Γ in the context of operator algebras. Let L(Γ) be the von Neumann algebra generated by the left regular representation of Γ. Let M be a finite factor and let U(M) be its unitary group. Let π:Γ→U(M) be a group homomorphism such that π(Γ)”=M. Then either (i) M is finite dimensional, or (ii) there exists a subgroup of finite index Λ of Γ such that π|Λ extends to a homomorphism U(L(Λ))→U(M). This answers, in the special case of SL n (ℤ), a question of A. Connes discussed in [Jone00, p. 86]. The result is deduced from a complete description of the tracial states on the full C *–algebra of Γ. As another application, we show that the full C *–algebra of Γ has no faithful tracial state, thus answering a question of E. Kirchberg.

Harish–Chandra homomorphisms and symplectic reflection algebras for wreath-products

Abstract: The main result of the paper is a natural construction of the spherical subalgebra in a symplectic reflection algebra associated with a wreath-product in terms of quantum hamiltonian reduction of an algebra of differential operators on a representation space of an extended Dynkin quiver. The existence of such a construction has been conjectured in [EG]. We also present a new approach to reflection functors and shift functors for generalized preprojective algebras and symplectic reflection algebras associated with wreath-products.

Conformal Designs based on Vertex Operator Algebras

Abstract: We introduce the notion of a conformal design based on a vertex operator algebra. This notation is a natural analog of the notion of block designs or spherical designs when the elements of the design are based on self-orthogonal binary codes or integral lattices, respectively. It is shown that the subspaces of fixed degree of an extremal self-dual vertex operator algebra form conformal 11-, 7-, or 3-designs, generalizing similar results of Assmus-Mattson and Venkov for extremal doubly-even codes and extremal even lattices. Other examples are coming from group actions on vertex operator algebras, the case studied first by Matsuo. The classification of conformal 6- and 8-designs is investigated. Again, our results are analogous to similar results for codes and lattices.

Operational Axioms for Quantum Mechanics

Abstract: The mathematical formulation of Quantum Mechanics in terms of complex Hilbert space is derived for finite dimensions, starting from a general definition of physical experiment and from five simple Postulates concerning experimental accessibility and simplicity. For the infinite dimensional case, on the other hand, a C*‐algebra representation of physical transformations is derived, starting from just four of the five Postulates via a Gelfand‐Naimark‐Segal (GNS) construction. The present paper simplifies and sharpens the previous derivation in Ref. [1]. The main ingredient of the axiomatization is the postulated existence of faithful states that allows one to calibrate the experimental apparatus. Such notion is at the basis of the operational definitions of the scalar product and of the transposed of a physical transformation. What is new in the present paper with respect to Ref. [1], is the operational deduction of an involution corresponding to the complex‐conjugation for effects, whose extension to transformations allows to define the adjoint of a transformation when the extension is composition‐preserving. The existence of such composition‐preserving extension among possible extensions is analyzed.