Positive integrable elements relative to a left Hilbert algebra

Let $\text{U}$ be an achieved left Hilbert algebra. Let $\text{η∈D}{}^{\mathrm{?}}$ be an element such that π′(η) is a positive operator. Then, following M. A. Rieffel, η is called integrable if sup{(η|e)e∈U and ee^?e²} < + ∞. It is shown that η is integrable if and only if there is an element ζ∈D^flat; such π′(ζ) is positive and ζ is a square root of η in an appropriate sense. This is shown to be a generalization of Godement's well known theorem on the existence of a convolution square root for a continuous square-integrable positive-definite function on a locally compact group. An “integral” and an “L¹-norm” are then defined on the linear span of the positive integrable elements and the completion of this space, denoted by L¹($\text{U}$), is shown to be the predual of $\text{l}$($\text{U}$). “Godement's theorem” is then used to investigate square-integrable representations of $\text{U}$.     

Positive semi-definiteness in the complex group algebra

Necessary and sufficient conditions for an element in the complex group algebra to be p.s.d. and properties of p.s.d. elements are given with applications to generalized matrix functions.     

Capelli elements of the group algebra

Inspired by the Capelli-type identities for group determinants researched by Tôru Umeda, we give Capelli identities for irreducible representations of any finite group, and Capelli elements of the group algebra associated with these identities. These elements construct a basis of the centre of the group algebra.     

Primitive elements in the matroid-minor Hopf algebra

We introduce the matroid-minor coalgebra C, which has labeled matroids as distinguished basis and coproduct given by splitting a matroid into a submatroid and complementary contraction in all possible ways. We introduce two new bases for C; the first of these is related to the distinguished basis by Möbius inversion over the rank-preserving weak order on matroids, the second by Möbius inversion over the suborder excluding matroids that are irreducible with respect to the free product operation. We show that the subset of each of these bases corresponding to the set of irreducible matroids is a basis for the subspace of primitive elements of C. Projecting C onto the matroid-minor Hopf algebra H, we obtain bases for the subspace of primitive elements of H.     

On the Lie algebra of skew-symmetric elements of an enveloping algebra

Let L be a restricted Lie algebra over a field of characteristic p>2 and denote by u(L) its restricted enveloping algebra. We establish when the Lie algebra of skew-symmetric elements of u(L) under the principal involution is solvable, nilpotent, or satisfies an Engel condition.     

A quantum octonion algebra

Using the natural irreducible 8-dimensional representation and the two spin representations of the quantum group (D) of D, we construct a quantum analogue of the split octonions and study its properties. We prove that the quantum octonion algebra satisfies the q-Principle of Local Triality and has a nondegenerate bilinear form which satisfies a q-version of the composition property. By its construction, the quantum octonion algebra is a nonassociative algebra with a Yang-Baxter operator action coming from the R-matrix of (D). The product in the quantum octonions is a (D)-module homomorphism. Using that, we prove identities for the quantum octonions, and as a consequence, obtain at new ``representation theory' proofs for very well-known identities satisfied by the octonions. In the process of constructing the quantum octonions we introduce an algebra which is a q-analogue of the 8-dimensional para-Hurwitz algebra.

     

Universal algebra of a Hom-Lie algebra and group-like elements

We construct the universal enveloping algebra of a Hom-Lie algebra and endow it with a Hom-Hopf algebra structure. We discuss group-like elements that we see as a Hom-group integrating the initial Hom-Lie algebra.     

On some issues related to the moment problem for the band matrices with operator elements

The connection between the classical moment problem and the spectral theory of second order difference operators (or Jacobi matrices) is a thoroughly studied topic. Here we examine a similar connection in the case of the second order operator replaced by an operator generated by an infinite band matrix with operator elements. For such operators, we obtain an analog of the Stone theorem and consider the inverse spectral problem which amounts to restoring the operator from the moment sequence of its Weyl matrix. We establish the solvability criterion for such problems, find the conditions ensuring that the elements of the moment sequence admit an integral representation with respect to an operator valued measure and discuss an algorithm for the recovery of the operator. We also indicate a connection between the inverse problem method and the Hermite-Padé approximations.     

Lifting problem of the measure algebra

We prove the consistency of “?/I _mz does not split” (see Notation). We write the proof so that with the standard duality, also the consistency of “?/I _tc does not split” (i.e., replacing measure zero by first category, random by generic, etc.) is proved. The method is the oracle chain condition.     

Local moment problem

The work is devoted to the local moment problem, which consists in finding of non-decreasing functions on the real axis having given first 2n + 1, n ≥ 0, power moments on the whole axis and also 2m + 1 first power moments on a certain finite axis interval. Considering the local moment problem as a combination of the Hausdorff and Hamburger truncated moment problems we obtain the conditions of its solvability and describe the class of its solutions with minimal number of growth points if the problem is solvable. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A twisted quantum toroidal algebra

As an analog of the quantum TKK algebra, a twisted quantum toroidal algebra of type A ₁ is introduced. Explicit realization of the new quantum TKK algebra is constructed with the help of twisted quantum vertex operators over a Fock space.     

On the second moment in Blachke's problem

Translated from Lietuvos Matematikos Rinkinys, Vol. 34, No. 2, pp. 149–154, April–June, 1994.     

Gaussian elements of a semicontent algebra

The connection between a univariate polynomial having locally principal content and the content function acting like a homomorphism (the so-called Gaussian property) has been explored by many authors. In this work, we extend several such results to the contexts of multivariate polynomials, power series over a Noetherian ring, and base change of affine K-algebras by separable algebraically closed field extensions. We do so by using the framework of the Ohm–Rush content function. The correspondence is particularly strong in cases where the base ring is approximately Gorenstein or the element of the target ring is regular.     

An algebra of skew primitive elements

We study the various term operations on the set of skew primitive elements of Hopf algebras, generated by skew primitive semi-invariants of an Abelian group of grouplike elements. All 1-linear binary operations are described and trilinear and quadrilinear operations are given a detailed treatment. Necessary and sufficient conditions for the existence of multilinear operations are specified in terms of the property of particular polynomials being linearly dependent and of one arithmetic condition. We dub the conjecture that this condition implies, in fact, the linear dependence of the polynomials in question and so is itself sufficient. Supported by RFFR grant No. 95-01-01356, and by the National Society of researchers, Mexico (exp. 18740, 1997). Translated fromAlgebra i Logika, Vol. 37, No. 2, pp. 181–223, March–April, 1998.     

Spectral properties of descent algebra elements

In this article, we determine the eigenvalues and their corresponding multiplicities of the action on the group algebra of a finite Coxeter group of an element of its descent algebra. Meanwhile, we identify a slight error in the paper of Bergeron, Bergeron, Howlett, and Taylor in a formula for certain structure coefficients in the descent algebra. We provide the correct formula, and give an example which explicitly shows the difference between both formulas.     

Carleman problem in a banach algebra

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 3, pp. 419–423, March, 1989.     

On the fourth moment in the Rankin-Selberg problem

If

The p-adic quantum plane algebras and quantum Weyl algebra

Let q be a principal unit of the ring of valuation of a complete valued field K, extension of the field of p-adic numbers. Generalizing Mahler basis, K. Conrad has constructed orthonormal basis, depending on q, of the space of continuous functions on the ring of p-adic integers with values in K. Attached to q there are two models of the quantum plane and a model of the quantum Weyl algebra, as algebras of bounded linear operators on the space of p-adic continuous functions. For q not a root of unit, interesting orthonormal (orthogonal) families of these algebras are exhibited and providing p-adic completion of quantum plane and quantum Weyl algebras. The text was submitted by the authors in English.