Some extensions of the Hewitt-Savage zero-one law

Summary Let denote the class of infinite product probability measures = ₁× ₂× defined on an infinite product of replications of a given measurable space (X, A), and let denote the subset of for which (A) =0 or 1 for each permutation invariant event A. Previous works by Hewitt and Savage, Horn and Schach, Blum and Pathak, and Sendler (referenced in the paper) discuss very restrictive sufficient conditions under which a given member , of belongs to . In the present paper, the class is shown to possess several closure properties. E.g., if and _{0 n} for some n 1, then ₀× ₁× ₂×.... While the current results do not permit a complete characterization of they demonstrate conclusively that is a much larger subset of than previous results indicated. The interesting special case X={0,1} is discussed in detail.Research supported by the National Science Foundation under grant No. MCS75-07556     

Simplicial cells in arrangements of hyperplanes

Let be an arrangement of n hyperplanes in P ^d , C() its cell complex, and H any hyperplane of . It is proved: (1) If is not a near pencil then there are at least n–d–1 simplicial d-cells of C(), each having no facet in H. (2) There are at least d+1 simplicial d-cells of C(), each having a facet in H.Material for this paper was taken from the author's doctoral dissertation.     

Studies on the Painlevé equations

Summary In this series of papers, we study birational canonical transformations of the Painlevé system , that is, the Hamiltonian system associated with the Painlevé differential equations. We consider also -function related to and particular solutions of . The present article concerns the sixth Painlevé equation. By giving the explicit forms of the canonical transformations of associated with the affine transformations of the space of parameters of , we obtain the non-linear representation: GG_*, of the affine Weyl group of the exceptional root system of the type F₄ A canonical transformation of G_* can extend to the correspondence of the -functions related to . We show the certain sequence of -functions satisfies the equation of the Toda lattice. Solutions of , which can be written by the use of the hypergeometric functions, are studied in details.     

Stop times in fock space stochastic calculus

Summary A stop time S in the boson Fock space over L ₂()₊ is a spectral measure in [0,] such that {S([0,t])} is an adapted process. Following the ideas of Hudson [6], to each stop time S a canonical shift operator U ^Sis constructed in . When S({}) has the vacuum as a null vector U ^Sbecomes an isometry. When S({})=0 it is shown that admits a factorisation _S]{S where _{S is the range of U ^Sand _S] is a suitable subspace of called the Fock space upto time S. This, in particular, implies the strong Markov property of quantum Brownian motion in the boson as well as fermion sense and the Dynkin-Hunt property that the classical Brownian motion begins afresh at each stop time. The stopped Weyl and fermion processes are defined and their properties studied. A composition operation is introduced in the space of stop time to make it a semigroup. Stop time integrals are introduced and their properties constitute the basic tools for the subject.     

On the fractional matching polytope of a hypergraph

For a hypergraph andb:⁺ define Conjecture. There is a matching of such that For uniform andb constant this is the main theorem of [4]. Here we prove the conjecture if is uniform or intersecting, orb is constant.The research was done while the author visited the Department of Mathematics at Rutgers University. Research supported in part by the Hungarian National Science Foundation under grant No. 1812Supported in party by NSF and AFOSR grants and by a Sloan Research Fellowship     

A new duality theory for compact groups

Summary The Tannaka-Krein duality theory characterizes the category (G) of finite-dimensional, continuous, unitary representations of a compact group as a subcategory of the category of Hilbert spaces. We prove a more powerful result characterizing (G) as an abstract category: every strict symmetric monoidalC ^*-category with conjugates which has subobjects and direct sums and for which theC ^*-algebra of endomorphisms of the monoidal unit reduces to the complex numbers is isomorphic to a category (G) for a compact groupG unique up to isomorphism.Research supported by the Ministero della Pubblica Istruzione and CNR-GNAFA     

Infinitely divisible completely positive mappings

Summary We generalise the theory of infinitely divisible positive definite functions f:G on a group G to a theory of infinite divisibility for completely positive mappings : G() taking values in the algebra of bounded operators on some Hilbert space .We prove a structure theorem for normalised infinitely divisible completely positive mappings which shows that the mapping , its Stinespring representation and its Stinespring isometry are of type S (in the sense of Guichardet [Gui]). Furthermore, we prove that a completely positive mapping is infinitely divisible if and only if it is the exponential (as defined in this paper) of a hermitian conditionally completely positive mapping.     

Construction d'une paramétrixe pour des opérateurs différentiels hypoelliptiques marimaux sur un espace homogéne d'un groupe de Lie nilpotent gradué de rang 2

Soient G une alébre de Lie nilpotente stratifée de rang 2, une sous-algébre de G, _0, la représentation de G dans l'espace L ²( \ G) indiute par le caractére trivial C, P un opérateur homogène appartenant à l'algébre universelle enveloppante (complexifiée) U(G) tel que l'opérateur _0, (P) soit hypoelliptique maximal. Cet opérateur peut s'exprimer par une intégrale dépendant de la restriction du symbole p de P au sousensemble = G · décrit par les orbites des éléments de dans la représentation contragrédiente de G dans G ^*.Une algèbre de symboles définis sur est construite et permet de déterminer une paramétrixe de _0, (P); des résultats de réguralité de cet opérateur dans des espaces de Sobolev adaptés sont ensuite obtenus.     

On the almost everywhere convergence of ergodic averages for power-bounded operators on LP-subspaces

Let X be a closed subspace of L^P(), where is an arbitrary measure and 1A(n) and (n) denote the discrete ergodic averages and Hilbert transform truncates defined by U. We extend to this setting the -a. e. convergence criteria forA(n) and (n) which V. F. Gaposhkin and R. Jajte introduced for unitary operators on L²(). Our methods lift the setting from X to ^p, where classical harmonic analysis and interpolation can be applied to suitable square functions.     

Prime and Primitive Ideals in Graded Deformations of Algebraic Quantum Groups at Roots of Unity

Let G be a connected, simply connected complex semisimple Lie group of rank n. The deformations employed by Artin, Schelter and Tate, and Hodges, Levasseur and Toro can be applied to the single parameter quantizations, at roots of unity, of the Hopf algebra of regular functions on G. Each of the resulting complex multiparameter quantum groups F _,p [G] depends on both a suitable root of unity and an antisymmetric bicharacter p: Z ⁿ ×Z ⁿ C ^×. These quantizations differ significantly from their single parameter (root-of-unity) counterparts, and, in particular, may have infinite-dimensional irreducible representations. Our approach to F _,p [G] depends on a natural ×-action thereon, where is an n-torus, and our main result offers a classification of the primitive ideals: We use a multiparameter quantum Frobenius map to provide a bijection from (PrimF _,p [G])/× onto G/H×H, where H is a maximal torus of G. In the single parameter case, this bijection is a consequence of work by De Concini and Lyubashenko, and De Concini and Procesi; our results require their analysis. Our methods also exploit earlier work by Moeglin and Rentschler concerning actions of algebraic groups on complex Noetherian algebras. In contrast to generic quantizations of the coordinate ring of G, the primitive spectrum of F _,p [G] is not finitely stratified by the torus action.     

Numerically regular hereditary classes of combinatorial geometries

A hereditary class of combinatorial geometries (or simple matroids) is a collection of geometries closed under minors and direct sums. A geometry G in is extremal if no proper extension of G of the same rank is in . The size function h(n) of is defined by h(n)=max {|G|: G and rank(G)=n}, where |G| is the number of points in G. A hereditary class is numerically regular if for every extremal geometry G in , |G|=h (rank(G)). We determine all the numerically regular hereditary classes for which the set {h(n)–h(n–1): 1n<} of positive integers does not have an upper bound: they are all varieties. We also give several examples of numerically regular hereditary classes which are not varieties.Partially supported by a North Texas State University Faculty Research Grant.     

Invarianz gewisser Operatorenklassen unter nichtkommutativen ergodischen Mitteln

Let be a group of *-automorphisms on the algebra of bounded linear operators on a complex Hilbert space H. Then the strongly closed convex hull of the orbit of any compact operator under consists of compact operators. The same is true if one replaces compact by nuclear, Hilbert-Schmidt or positive Fredholm. We further discuss these results in the framework of the noncommutative mean ergodic theorem of KOVACS and SZ#x00FC;CS and formulate an analogous theorem for the algebra of compact operators on a complex Hilbert space.

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Gefördert von der Deutschen Forschungsgemeinschaft im Rahmen des Forschungsvorhabens Ko 506/1.     

When Is a Subspace of B(H) Ideal?

Let be a real or complex Hilbert space and let () denote the algebra of all bounded linear operators on . We show that if N is a subspace of (H) and for positive operators P₁, P₂ and every A N, P₁P₂A^* + AP₁P₂ N then N is an ideal. Furthermore if is an infinite dimensional real space then N = (H).AMS Subject Classification (1991): Primary 47B47, 47D25     

Tessellations generated by hyperplanes

Let be a locally finite system of hyperplanes in ^d with the property that the cells of the induced cell complex decomposition of ^d have uniformly bounded diameters. If is simple and the density of the vertices in exists, then the density of thek-cells in exists and can be given explicitly (k = 1, ...,d). Also, the mean number ofj-faces of thek-cells in exists and can be calculated. For certain nonsimple systems , corresponding inequalities are obtained.     

On positivity in Hecke algebras

Let be the Hecke algebra associated with a Coxeter system (W, R). The structure constants of with respect to various bases are Laurent polynomials, whose coefficients enjoy remarkable positivity properties. We survey these and prove some new ones using the relationship between and the geometry of Schubert varieties.To Professor Jacques Tits on his sixtieth birthday     

Convergence theorems for set-valued amarts and uniform amarts

( ) , — , - . .     

A free convenient vector space for holomorphic spaces

In this paper we show that there exists a free convenient vector space for the case of holomorphic spaces and holomorphic maps. This means that for every spaceX with a holomorphic structure, there exists an appropriately complete locally convex vector space X and a holomorphic mapl _X:XX, such that for any vector space of the same kind the map (l _X )^*:L(X,E)(X,E) is a bijection. Analogously to the smooth case treated in [2, 5.1.1] the free convenient vector space X can be obtained as the Mackey closure of the linear subspace spanned by the image of the canonical mapX(X).In the second part of the paper we prove that in the case whereX is a Riemann surface, one hasX=(X,).     

Multiplicity One for Representations Corresponding to Spherical Distribution Vectors of Class ρ

In this paper one considers a unimodular second countable locally compact group G and the homogeneous space X:=H\G, where H is a closed unimodular subgroup of G. Over X complex vector bundles are considered such that H acts on the fibers by a unitary representation with closed image. The natural action of G on the space of square integrable sections is unitary and possesses an integral decomposition in so-called spherical distributions of class . The uniqueness of this decomposition can be characterized by a number of equivalent properties. Uniqueness is shown to hold for a class of semidirect products. In the case that H is compact, the multiplicity free decomposition is shown to be equivalent with the commutativity of a suitable convolution algebra. As an example, one takes for X a symmetric k-variety , with k a locally compact field of characteristic not equal to two, and for a character of _k, whose square is trivial. Here is a reductive algebraic group defined over k and is the fixed point group of an involution of defined over k. It is shown then that the natural representation of G_k on the Hilbert space is multiplicity free if is anisotropic. Next a criterion is derived that leads to multiplicity one also in the noncompact situation. Finally, in the non-Archimedean case, a general procedure is given that might lead to showing that a pair is a generalized Gelfand pair. Here and are suitable algebraic groups defined over k.Mathematics Subject Classifications (2000) 20G15, 20G20, 22E15, 22E46.Aloysius G. Helminck: Partially supported by N.S.F. Grant DMS-9977392.     

A Comparison of Two Spaces of Generalized White Noise Functionals

In the literature (see [5, 6, 8]) there are two families of spaces called Kondratiev spaces: (_c)^± and (S _c)^± for 0 1. We investigate the relation between the spaces and show that they are topologically isomorphic when (^d) L2 (^d) (^d) is the underlying Gel'fand triple for (_c)^±. In this case we also give the explicit relation between the S-transform and -transform on (_c)^-1 and (S _c)^-1, respectively.     

On the fine structure of the polygroup blow-up

We study in detail the blow-up procedure described in [BTW01]. We obtain a structure theorem for coreless polygroups as a double quotient space G//H, and a polygroup chunk theorem. Seeking to remove the arbitrary parameter needed for the blow-up, we find canonical Ø-invariant groupoids analogous to G and H above, and show that contains precisely all the arbitrary choices related to the blow-up. Mathematics Subject Classification (2000): 03C45, 03C60