Positive elements in the algebra of the quantum moment problem

Summary Let denote the extended Weyl algebra, , the Weyl algebra. It is well known that every element of of the formA=B _k ^* B _k is positive. We prove that the converse implication also holds: Every positive elementA in has a quadratic sum factorization for some finite set of elements (B _k ) in . The corresponding result is not true for the subalgebra . We identify states on which do not extend to states on . It follows from a result of Powers (and Arveson) that such states on cannot be completely positive. Our theorem is based on a certain regularity property for the representations which are generated by states on , and this property is not in general shared by representations generated by states defined only on the subalgebra .Work supported in part by the NSF     

On the convergence of powers of interval matrices (2)

Summary Let be a real irreduciblen×n interval matrix. Then a necessary and sufficient condition is given for the sequence of the powers of an interval matrix to converge to a matrix which is not the null matrix. In addition a criterion for is proved to decide whether the limit matrix satisfies the condition of symmetry .     

On periodic groups of automorphisms of extremal groups

It is proved that if a periodic group has an extremal normal divisor , determining a complete abelian factor group , then the center of the group contains a complete abelian subgroup , satisfying the relation and intersecting on a finite subgroup. It is also established with the aid of this proposition that every periodic group of automorphisms of an extremal group is a finite extension of a contained in it subgroup of inner automorphisms of the group .Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp. 91–96, July, 1968.     

Kinematik in der Laguerre-Ebene I

The geometry of the real Laguerre-plane is defined as the Cayley-Klein-geometry of a 3-dimensional affine space with a non-degenerate absolute conic in the ideal plane. A one-parametric family of transformations of the Laguerre-plane is called a . In this paper the kinematical foundations of the -motions are developed. Furthermore there is given a linear construction of the tangents of the curves generated under a -motion. At the end the dual and the inverse -motions are considered.     

Generalized Harish-Chandra Modules: A New Direction in the Structure Theory of Representations

Let be a reductive Lie algebra over C. We say that a -module M is a generalized Harish-Chandra module if, for some subalgebra , M is locally -finite and has finite -multiplicities. We believe that the problem of classifying all irreducible generalized Harish-Chandra modules could be tractable. In this paper, we review the recent success with the case when is a Cartan subalgebra. We also review the recent determination of which reductive in subalgebras are essential to a classification. Finally, we present in detail the emerging picture for the case when is a principal 3-dimensional subalgebra.     

Kinematik in der Laguerre-Ebene II: 97-0197-0197-01-Bewegungen mit Polkurven

This second part of kinematics in the Laguerre-plane contains differential geometric properties of -motions. In this paper we consider only such -motions which are instantaneous products of a rotation and a homothetic transformation with common center. The centers, called poles of a -motion, generate a curve on a ruled surface consisting of fixed lines. First we investigate the dependence of a -motion on the pair of strips along the pole curves which contact each other. Furthermore the points with isotropic tangents of their paths lie on a cone at each time of a -motion. On this cone there exist curves of some interest like the curves of the cycles of inflection and of the vertical cycles and the Bresse-curves. The isotropic tangents of the paths in the cycles of inflection define a socalled hypercycle of Blaschke.     

Completion of an additive measure to a σ-additive measure by an extension of the space

For an algebra of subsets of a set X there is constructed a set and an algebra of its subsets so that the mapping is a one-to-one correspondence between and and for each additive measure on the measure on defined by the equation is countably additive.Translated from Matematicheskie Zametki, Vol. 3, No. 1, pp. 71–76, January, 1968.The author wishes to express his deep appreciation to S. V. Fomin, under whose guidance this paper was written.     

Chu-Dualität und zwei Klassen maximal fastperiodischer Gruppen

This is a continuation of the paper Zwei Klassen lokalkompakter maximal fastperiodischer Gruppen, [6]. In [6], the classes and were introduced. We give sufficient conditions to conclude thatG is in if one knows thatG/G ₀ is in . If a groupG is in and ifG satisfies the Chu-duality then all closed subgroups ofG satisfy the Chu-duality. The Chu-quasi-dual of the Heisenberg groupH with integral coefficients is computed. It is shown thatH does not satisfy the Chu-duality, thatH is in , and thatH is not in .     

Homological Dimensions of the Algebra Formed by Entire Functions of Elements of a Nilpotent Lie Algebra

This note deals with homological characteristics of algebras of holomorphic functions of noncommuting variables generated by a finite-dimensional nilpotent Lie algebra . It is proved that the embedding of the universal enveloping algebra of into its Arens–Michael hull is an absolute localization in the sense of Taylor provided that      

On the Spectrum of Degenerate Operator Equations

We study the distribution in the complex plane of the spectrum of the operator , generated by the closure in of the operation originally defined on smooth functions with values in a Hilbert space satisfying the Dirichlet conditions . Here and A is a model operator acting in . Criterial conditions on the parameter for the eigenfunctions of the operator to form a complete and minimal system as well as a Riesz basis in the Hilbert space H are given.     

Derangements and Tensor Powers of Adjoint Modules for \mathfrak{s}\mathfrak{l}_n

We obtain the decomposition of the tensor space as a module for , find an explicit formula for the multiplicities of its irreducible summands, and (when n 2k) describe the centralizer algebra = ( ) and its representations. The multiplicities of the irreducible summands are derangement numbers in several important instances, and the dimension of is given by the number of derangements of a set of 2k elements.     

Measurable Hermitian-positive functions

Let , respectively, denote the sets of continuous, measurable, and almost-everywhere vanishing functions f(x) (–a<x<a; f(0)>0). The theorem is proved that for every there correspond and , such that f=f_c + f_s. Some unsolved problems related to this theorem are formulated.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 79–89, January, 1978.     

Complétude des métriques lorentziennes de\mathbb{T}^2 et difféomorphismes du cercle

Let and the foliations by the null geodesics of some lorentzian metricg on the torus . We analyse how geodesic completeness properties ofg are related to the dynamics of and .     

A new product of algebras and a type reduction theorem

A construction is defined which associates, to every algebra of a fixed but arbitrary finite similarity type, a groupoidF . The identities ofF are finitely based if and only if those of are, andF is finite if and only if is finite. Up to isomorphism,F has the same endomorphism monoid and subalgebra lattice as , but the congruence lattice ofF is the result of adjoining a new 1 to the congruence lattice of .F is functorial, preserves the satisfaction (and the non-satisfaction) of most Mal'cev conditions, and produces, by composition with the operation of forming the generated variety, an isomorphism of the lattice of varieties of fixed type to an interval in the lattice of varieties of groupoids.The construction makes use of a new product operation, applicable to two algebras of differing similarity types, which is introduced and studied in this paper.Research supported by National Science Foundation grant MCS-8103455.Presented by K. A. Baker.     

Regular and singular Hermitian operators

Let A be a closed Hermitian operator, let be the orthogonal complement of the domain of definition of A, and let be the defect subspace. An operator A is called regular if the orthogonal projection of on is closed. Criteria for regularity are established.Translated from Matematicheskie Zametki, Vol. 8, No. 2, pp. 197–203, August, 1970.     

Bounded solutions for 119-01119-01119-01-problem in pseudo-Siegel domains

Summary We study the problem of the existence of bounded solutions for the equation on pseudo-Siegel domains when the data satisfies the condition .

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Sunto Si studia il problema dell'esistenza di soluzioni limitate per l'equazione sui domini pseudo-Siegel quando il dato soddisfa alla condizione .

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Work done within the Project 40% M.U.R.S.T. «Geometria Reale e Complessa».     

Hankel operators on the weighted Bergman spaces with exponential type weights

Let denote the closed subspace of consisting of analytic functions in the unit disk . For certain class of subharmonic , the Hankel operatorH _b on with symbol is studied. Criteria for boundedness and compactness of such kind of Hankel operators are presented.R. Rochberg's research was partially supported by a grant from the National Science Foundation.     

Isometric dilations in nest algebras

LetT be a contraction acting in a separable Hilbert space and leaving invariant a nest of subspaces of . We answer the question: when doesT have an isometric extension to which leaves invariant the nest = {N N :N ;}.     

Topological and bornological characterisations of ideals in von Neumann algebras: I

Suppose is a von Neumann algebra on a Hilbert space and is any ideal in . We determine a topology on , for which the members of that are to norm continuous are exactly those in ; and a bornology on such that the elements of which map the unit ball to an element of , equivalently those members of that are norm to bounded, are exactly those in . This is achieved via analogues of the notions of injectivity and surjectivity in the theory of operator ideals on Banach spaces.