Some Examples of Graded C*-Algebras

We apply the theory of C*-algebras graded by a semilattice to crossed products of C*-algebras. We establish a correspondence between the spectrum of commutative graded C*-algebras and the spectrum of their components. This will allow us to compute the spectrum of some commutative examples of graded C*-algebras.     

A Topos for Algebraic Quantum Theory

The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C*-algebra of observables A induces a topos \({\mathcal{T}(A)}\) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra \({\underline{A}}\) . According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum \({\underline{\Sigma}(\underline{A})}\) in \({\mathcal{T}(A)}\) , which in our approach plays the role of the quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on \({\underline{\Sigma}}\) , and self-adjoint elements of A define continuous functions (more precisely, locale maps) from \({\underline{\Sigma}}\) to Scott’s interval domain. Noting that open subsets of \({\underline{\Sigma}(\underline{A})}\) correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos \({\mathcal{T}(A)}\).These results were inspired by the topos-theoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.     

On the integral representation of states on a C*-algebra

The purpose of this paper is to give some complements to the various extremal decompositions of states on aC*-dynamical system i.e. a pair (A, G) whereA is aC*-algebra andG is a group acting on aA by *-automorphisms. We shall see for instance that the method of decomposition associated with a maximal abelianW*-algebra does not give all the extremal measures in the general case. We also give the explicit form of the greatest lower bound of all the extremal measures and a certain form of continuity of the decomposition. Finally we characterize various systems in the literature (G-abelian algebras, large systems and quasi-large systems) in terms of the equivalence of different notions of ergodicity.     

Real Structures on Almost-Commutative Spectral Triples

We refine the reconstruction theorem for almost-commutative spectral triples to a result for real almost-commutative spectral triples, clarifying in the process both concrete and abstract definitions of real commutative and almost-commutative spectral triples. In particular, we find that a real almost-commutative spectral triple algebraically encodes the commutative *-algebra of the base manifold in a canonical way, and that a compact oriented Riemannian manifold admits real (almost-)commutative spectral triples of arbitrary KO-dimension. Moreover, we define a notion of smooth family of real finite spectral triples and of the twisting of a concrete real commutative spectral triple by such a family, with interesting KK-theoretic and gauge-theoretic implications.     

Endomorphisms of Quantum Generalized Weyl Algebras

We prove that every endomorphism of a simple quantum generalized Weyl algebra A over a commutative Laurent polynomial ring in one variable is an automorphism. This is achieved by obtaining an explicit classification of all endomorphisms of A. Our main result applies to minimal primitive factors of the quantized enveloping algebra \({U_q({\mathfrak{sl}}_2)}\) and certain minimal primitive quotients of the positive part of \({U_q({\mathfrak{so}}_5)}\) .     

On Pythagoras Theorem for Products of Spectral Triples

We discuss a version of Pythagoras theorem in noncommutative geometry. Usual Pythagoras theorem can be formulated in terms of Connes’ distance, between pure states, in the product of commutative spectral triples. We investigate the generalization to both non-pure states and arbitrary spectral triples. We show that Pythagoras theorem is replaced by some Pythagoras inequalities, that we prove for the product of arbitrary (i.e. non-necessarily commutative) spectral triples, assuming only some unitality condition. We show that these inequalities are optimal, and we provide non-unital counter-examples inspired by K-homology.     

Multiplicities in highp t hadron-nucleus interactions

We present results obtained in π^? A (A=C, Cu, Pb) — collisions at 38 GeV/c. A single particle trigger selects events with one charged particle in the central region and large transverse momentum. The effect of this trigger on the multiplicities of all charged particles and of protons is shown.     

Source Identities and Kernel Functions for Deformed (Quantum) Ruijsenaars Models

We consider the relativistic generalization of the quantum A _N-1 Calogero–Sutherland models due to Ruijsenaars, comprising the rational, hyperbolic, trigonometric and elliptic cases. For each of these cases, we find an exact common eigenfunction for a generalization of Ruijsenaars analytic difference operators that gives, as special cases, many different kernel functions; in particular, we find kernel functions for Chalykh–Feigin–Veselov–Sergeev-type deformations of such difference operators which generalize known kernel functions for the Ruijsenaars models. We also discuss possible applications of our results.     

Comment on the surface exponential for tensor fields

Starting from an essentially commutative exponential map E(B|I) for generic tensor-valued 2-forms B, which were introduced in [10] as a direct generalization of the ordinary noncommutative P exponent for 1 forms with values in matrices (i.e., in tensors of rank 2), we suggest a nontrivial but multiparametric exponential E(B|I|t_γ), which can serve as an interesting multidirectional evolution operator in the case of higher ranks. To emphasize the most important aspects of the article, the construction is restricted to the backgrounds I _ijk , which are associated with the structure constants of the commutative associative algebras, which make it insensitive to the topology of the 2D surface. Boundary effects are also eliminated (straightforward generalization is needed to incorporate them).     

Etude spectrale d'une famille d'opérateurs non-symétriques intervenant dans la théorie des champs de reggeons

In this paper, we study a few spectral properties of a non-symmetrical operator arising in the Gribov theory. The first and second section are devoted to Bargmann's representation and the study of general spectral properties of the operator: $$\begin{gathered} H_{\lambda ',\mu ,\lambda ,\alpha } = \lambda '\sum\limits_{j = 1}^N {A_j^{ * 2} A_j^2 + \mu \sum\limits_{j = 1}^N {A_j^ * A_j + i\lambda \sum\limits_{j = 1}^N {A_j^ * (A_j + A_j^ * )A_j } } } \hfill \\ + \alpha \sum\limits_{j = 1}^{N - 1} {(A_{j + 1}^ * A_j + A_j^ * A_{j + 1} ),} \hfill \\ \end{gathered}$$ whereA* _j andA _j ,j∈[1,N] are the creation and annihilation operators. In the third section, we restrict our study to the case of nul transverse dimension (N=1). Following the study done in [1], we consider the operator: $$H_{\lambda ',\mu ,\lambda } = \lambda 'A^{ * 2} A^2 + \mu A^ * A + i\lambda A^ * (A + A^ * )A,$$ whereA* andA are the creation and annihilation operators. For λ′>0 and λ′²≦μλ′+λ². We prove that the solutions of the equationu′(t)+H _{λ′, μ,λ} u(t)=0 are expandable in series of the eigenvectors ofH _λ′,μ,λ fort>0. In the last section, we show that the smallest eigenvalue σ(α) of the operatorH _{λ′,μ,λ,α} is analytic in α, and thus admits an expansion: σ(α)=σ₀+ασ₁+α²σ₂+..., where σ₀ is the smallest eigenvalue of the operatorH _{λ′,μ,λ,0}.     

On a generalization of Wick-powers for generalized free-fields in terms of Jacobi-fields

Within the frame of Jacobi-fields a generalization of Wick-powers of generalized free fields is proposed. The key notion is that of a (generalized) contraction map. Those contraction maps F which yield a relativistic quantum field A^F are characterized. Using some simplifying assumptions the general form of a contraction map F which yields a relativistic quantum field A^F is determined. Furthermore, those contraction maps F are characterized for which A^F is in the Borchers class of the generalized free field A we start with. The Wick-product of generalized free fields appears as a particular example of this construction.     

Positive cones in enveloping algebras

The concept of strong ordering on enveloping algebras of finite-dimensional Lie algebras is introduced and studied as a generalization of the corresponding notion for the commutative polynomial algebra. A linear functional f on an enveloping algebra $\text{E}$ (G) is called strongly positive if f(x) ? 0 for all x ? $\text{E}$(G) which are mapped on positive operators for all G-integrable irreducible representations of $\text{E}$(G). We prove that for each real connected Lie group G ≠ R₁ there are positive, not strongly positive, linear functionals on $\text{E}$(G). A non-commutative problem of moments is defined. It has a solution iff the corresponding linear functional is strongly positive.     

Emission characteristics of a barrier discharge in an argon-freon-water vapor mixture in the UV-VUV spectral range

Optical characteristics of an ArCl*-OH* lamp excited by a nanosecond barrier discharge are studied. This discharge is a source of the ArCl (B → X), (D’ → A’), and OH(A → X) molecular band emission with peaks at 175, 258, and 309 nm, respectively. The intensity of the barrier discharge plasma radiation is optimized as a function of the CCl₄ vapor partial pressure at p(Ar) = 24 kPa and p(H₂O) = 10–20 Pa.     

Generalizing some properties of short range classical solutions to Yang-Mills theory

We extend a theorem which states that for classical solutions of Yang-Mills theory, the field G_μν has to decrease at least as fast as the source S_μ at spatial infinity, provided G_μν decreases exponentially [G_μν ～ exp(?Mr)]. This generalization encompasses all decreases G_μν ～ exp(?Mr^η) with η > 0, r→∞. This is done by assuming an integral representation for A_μ, the vector potential, and imposing some regularity conditions on A_μ, valid as r→∞.     

Sur une Valeur propre d'un Operateur

The studied eigenvalue is the smallest one of the operator: $$H_\mu = \mu A* + i\lambda A*(A + A*)A$$ in the orthogonal complement of the vacuum whereA* andA are the creation and annihilation operators. Call this eigenvalueE(μ) as attention is focused on the dependence on μ. This eigenvalue exists and is a positive number for positive values of μ. Using the fact that the inverse ofH _μ is a positive operator, it is proved thatE extends to a positive, increasing, analytic function on the whole real line. In particular,E(0)≠0, contrary to what might have been expected from the fact thatA*(A+A*)A is formally self-adjoint.     

Invariant states and conditional expectations of the anticommutation relations

The groupG of unitary elements of a maximal abelian von Neumann algebra on a separable, complex Hilbert spaceH acts as a group of automorphisms on the CAR algebraA(H) overH. It is shown that the set ofG-invariant states is a simplex, isomorphic to the set of regular probability measures on aw*-compact setS ofG-invariant generalized free states. The GNS Hilbert space induced by an arbitraryG-invariant state onA(H) supports a *-representation ofC(S); the canonical map ofA(H) intoC(S) can then be locally implemented by a normal,G-invariant conditional expectation.     

Extensions of Hilbert C*-modules: Classification in simple cases

The development of the theory of extensions of Hilbert C*-modules was initiated by Baki? and Guljas?. An easy observation shows that, if the underlying C*-algebra extension is commutative and the Hilbert C*-modules are projective of finite type, then the algebraic properties of the corresponding Busby invariant enable one to identify extensions with isometric mappings of the corresponding vector bundles. For the case in which the Hilbert C*-modules are free of rank one, we evaluate the set of extensions in topological terms.