Wick products of the CCR algebra

The purpose of this paper is to put in a precise mathematical (algebraic) form the Wick products of the CCR algebra. We state in detail the reduction of ordinary product of Bose fields in terms of a finite sum of monomials in the creation and annihilation operators in which all creation operators occur to the left of all annihilation operators (Wick‐ordered) and the Fock (vacuum) state of the former. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Local Minimal Curves in Homogeneous Reductive Spaces of the Unitary Group of a Finite von Neumann Algebra

We study the metric geometry of homogeneous reductive spaces of the unitary group of a finite von Neumann algebra with a non complete Riemannian metric. The main result gives an abstract sufficient condition in order that the geodesics of the Levi-Civita connection are locally minimal. Then, we show how this result applies to several examples. To my family     

Estimates for Littlewood-Paley Functions and Extrapolation

We prove certain L ^p -estimates for Littlewood-Paley functions arising from rough kernels. The estimates are useful for extrapolation to prove L ^p -boundedness of the Littlewood-Paley functions under a sharp kernel condition.      

Normal forms for real quadratic forms

We investigate the non-diagonal normal forms of a quadratic form on , in particular for n = 3. For this case it is shown that the set of normal forms is the closure of a 5-dimensional submanifold in the 6-dimensional Grassmannian of 2-dimensional subspaces of . Received: 27 June 2008     

Unique solvability for a class of full non-Newtonian fluids of one dimension with vacuum

We prove the local existence and uniqueness of the strong solutions for a class of full non-Newtonian fluids in one space dimension with the hypotheses that the initial data are small in some sense and satisfy some compatibility conditions. The initial density need not be positive, which means that we allow the initial vacuum.     

Finite-Rank Products of Toeplitz Operators in Several Complex Variables

For any α > −1, let A²_α be the weighted Bergman space on the unit ball corresponding to the weight (1 – |z|²)^α. We show that if all except possibly one of the Toeplitz operators are diagonal with respect to the standard orthonormal basis of A²_α and has finite rank, then one of the functions must be the zero function.      

Spectral Radius Algebras of Idempotents

We show that the spectral radius algebras of certain quadratic operators possess nontrivial invariant subspaces. Additionally, such algebras properly contain the operator’s commutant, so that the invariant subspaces are in some sense beyond hyperinvariant. The spectral radius algebras of idempotents are completely described and, as a consequence, it is shown that every intransitive collection of operators must be contained in a norm-closed proper spectral radius algebra.      

Hardy-Littlewood system and representations of integers by an invariant polynomial

Let f be an integral homogeneous polynomial of degree d, and let be the level set for each . For a compact subset in ), set

Infinite Classes of Dihedral Snarks

Flower snarks and Goldberg snarks are two infinite families of cyclically 5–edge–connected cubic graphs with girth at least five and chromatic index four. For any odd integer k, k > 3, there is a Flower snark, say J _k , of order 4k and a Goldberg snark, say B _k , of order 8k. We determine the automorphism groups of J _k and B _k for every k and prove that they are isomorphic to the dihedral group D _4k of order 4k. Research performed within the activity of INdAM–GNSAGA with the financial support of the Italian Ministry MIUR, project “Strutture Geometriche, Combinatoria e loro Applicazioni”.     

On a conjecture of Schmutz

In this work we discuss Schmutz’s conjecture that in dimension 2 to 8 the distinct norms that occur in the lattices with the best known sphere packings are strictly greater than those in any other lattice of the same covolume. We see that the ternary conjecture is not true. However, it seems that there is but one exception: one lattice, where for one length the conjecture fails. Received: 11 February 2008, Revised: 20 May 2008     

On quadratic fields generated by polynomials

Let be a polynomial of degree d ≥ 2 without multiple roots. Under the assumption of the ABC-conjecture, an asymptotic formula for the number of distinct fields among for has recently been given by Cutter, Granville, and Tucker. We use bounds for character sums to obtain an unconditional lower bound on the number of such fields for . Received: 19 November 2007     

A Clifford Algebraic Framework for $$\mathfrak{sp}(m)$$-Invariant Differential Operators

We introduce a framework for studying differential operators which are invariant with respect to the real (complex) symplectic Lie algebra ( ), associated to a quaternionic structure on a vector space . To do so, these algebras are realized within the orthogonal Lie algebra . This leads in a natural way to a refinement of the recently introduced notion of complex Hermitean Clifford analysis, in which four variations of the classical Dirac operator play a dominant role. David Eelbode: Postdoctoral fellow supported by the F.W.O. Vlaanderen (Belgium).     

Maximal subnear-rings of functions

For a group G, let M(G) denote the near-ring of functions on G. We characterize all maximal subnear-rings of M(G) and show that for many classes of groups, E(G), the near-ring generated by the semigroup, End(G) of G, is never maximal as a subnear-ring of M ₀ (G). Received: 25 April 2008     

Distance sets corresponding to convex bodies

Suppose that is a 0-symmetric convex body which denes the usual norm

Quadrature Domains with Infinite Volume

We discuss quadrature domains for subharmonic functions and prove the existence of core quadrature domains for certain positive measures. The core quadrature domains are the smallest quadrature domains as measures and inherit good properties from quadrature domains with finite volume. We next discuss new balayage for the class of harmonic functions integrable in a neighborhood of ∞. We give several estimates of balayage measures. The new balayage is introduced to construct quadrature domains for harmonic functions. Submitted: June 26, 2008. Accepted: July 24, 2008.     

A p-adic analogue of a formula of Ramanujan

During his lifetime, Ramanujan provided many formulae relating binomial sums to special values of the Gamma function. Based on numerical computations, Van Hamme recently conjectured p-adic analogues to such formulae. Using a combination of ordinary and Gaussian hypergeometric series, we prove one of these conjectures. Received: 11 April 2008     

Metric Properties of Projections in Semi-Hilbertian Spaces

Several results on norms of projections on a Hilbert space are extended for the operator seminorm defined by a positive semidefinite operator . The authors were supported in part by UBACYT I030, ANPCYT PICT 03-09521, PIP 2188/00.     

The Howe Duality for the Dunkl Version of the Dirac Operator

The classical Fischer decomposition of polynomials on Euclidean space makes it possible to express any polynomial as a sum of harmonic polynomials multiplied by powers of |x|². A deformation of the Laplace operator was recently introduced by Ch.F. Dunkl. It has the property that the symmetry with respect to the orthogonal group is broken to a finite subgroup generated by reflections (a Coxeter group). It was shown by B. ?rsted and S. Ben Said that there is a deformation of the Fischer decomposition for polynomials with respect to the Dunkl harmonic functions. In Clifford analysis, a Dunkl version of the Dirac operator was introduced and studied by P. Cerejeiras, U. K?hler and G. Ren. The aim of the article is to describe an analogue of the Fischer decomposition for solutions of the Dunkl Dirac operator. The main methods used are coming from representation theory, in particular, from ideas connected with Howe dual pairs. This paper is dedicated to the memory of our friend and colleague Jarolim Bureš to recall our long lasting cooperation on many topics in mathematics The work presented here is a part of the research project MSM 0021620839 and was supported also by the grant GA ČR 201/05/2117. Received: October, 2007. Accepted: February, 2008.     

Pencils on double coverings of curves

Let X be a smooth curve of genus g. When and d ≥ π−2g+1 we show the existence of a double covering where C a smooth curve of genus π with a base-point-free pencil of degree d which is not the pull-back of a pencil on X. Received: 7 February 2007; Revised: 1 July 2008     

Characterization of Amalgamated Free Blocks of a Graph von Neumann Algebra

We identify two noncommutative structures naturally associated with countable directed graphs. They are formulated in the language of operators on Hilbert spaces. If G is a countable directed graphs with its vertex set V(G) and its edge set E(G), then we associate partial isometries to the edges in E(G) and projections to the vertices in V(G). We construct a corresponding von Neumann algebra as a groupoid crossed product algebra of an arbitrary fixed von Neumann algebra M and the graph groupoid induced by G, via a graph-representation (or a groupoid action) α. Graph groupoids are well-determined (categorial) groupoids. The graph groupoid of G has its binary operation, called admissibility. This has concrete local parts , for all e ∈ E(G). We characterize of , induced by the local parts of , for all e ∈ E(G). We then characterize all amalgamated free blocks of . They are chracterized by well-known von Neumann algebras: the classical group crossed product algebras , and certain subalgebras (M) of operator-valued matricial algebra . This shows that graph von Neumann algebras identify the key properties of graph groupoids. Received: December 20, 2006. Revised: March 07, 2007. Accepted: March 13, 2007.