A description of Hilbert -modules in which all closed submodules are orthogonally closed

Let , be -algebras and a full Hilbert --bimodule such that every closed right submodule is orthogonally closed, i.e., . Then there are families of Hilbert spaces , such that and are isomorphic to -direct sums , resp. , and is isomorphic to the outer direct sum .

     

Type factors generated by purely infinite simple C*-algebras associated with free groups

Let be a free product of at least two but at most countably many cyclic groups. With each such group we associate a family of C*-algebras, denoted and generated by the reduced group C*-algebra and a collection of projections onto the -spaces over certain subsets of . We determine , the weak closure of in , and use this result to show that many of the C*-algebras in question are non-nuclear.

     

The Wiener transform on the Besicovitch spaces

In his fundamental research on generalized harmonic analysis, Wiener proved that the integrated Fourier transform defined by is an isometry from a nonlinear space of functions of bounded average quadratic power into a nonlinear space of functions of bounded quadratic variation. We consider this Wiener transform on the larger, linear, Besicovitch spaces defined by the norm . We prove that maps continuously into the homogeneous Besov space for and , and is a topological isomorphism when .

     

The maximal ideal space of with respect to the Hadamard product

It is shown that the space of all regular maximal ideals in the Banach algebra with respect to the Hadamard product is isomorphic to The multiplicative functionals are exactly the evaluations at the -th Taylor coefficient. It is a consequence that for a given function in and for a function holomorphic in a neighborhood of with and for all the function is in

     

On the divergence of the means of double Walsh-Fourier series

In 1992, Móricz, Schipp and Wade proved the a.e. convergence of the double means of the Walsh-Fourier series () for functions in ( is the unit square). This paper aims to demonstrate the sharpness of this result. Namely, we prove that for all measurable function we have a function such as and does not converge to a.e. (in the Pringsheim sense).

     

The Furuta inequality in Banach -algebras

Let be real numbers with and Furuta (1987) proved that if bounded linear operators on a Hilbert space satisfy , then . This inequality is called the Furuta inequality and has many applications. In this paper, we prove that the Furuta inequality holds in a unital hermitian Banach -algebra with continuous involution.

     

Refinable subspaces of a refinable space

Local refinable finitely generated shift-invariant spaces play a significant role in many areas of approximation theory and geometric design. In this paper we present a new approach to the construction of such spaces. We begin with a refinable function which is supported on . We are interested in spaces generated by a function built from the shifts of .

     

-semigroups generated by second order differential operators with general Wentzell boundary conditions

Let us consider the operator where is positive and continuous in and is equipped with the so-called generalized Wentzell boundary condition which is of the form at each boundary point, where This class of boundary conditions strictly includes Dirichlet, Neumann and Robin conditions.

Under suitable assumptions on , we prove that generates a positive -semigroup on and, hence, many previous (linear or nonlinear) results are extended substantially.

     

-sequences

A sequence of positive integers is called a -sequence if every integer has at most representations with all in and . A -sequence is also called a -sequence or Sidon sequence. The main result is the following

Theorem. Let be a -sequence and for an integer . Then there is a -sequence of size , where .

Corollary. Let . The interval then contains a -sequence of size , when .

     

Remarks about Schlumprecht space

Let denote the Schlumprecht space. We prove that

(1) is finitely disjointly representable in ;

(2) contains an -spreading model;

(3) for any sequence of natural numbers, is isomorphic to the space .

     

Periodic hyperfunctions and Fourier series

Every periodic hyperfunction is a bounded hyperfunction and can be represented as an infinite sum of derivatives of bounded continuous periodic functions. Also, Fourier coefficients of periodic hyperfunctions are of infra-exponential growth in , i.e., for every and every . This is a natural generalization of the polynomial growth of the Fourier coefficients of distributions.

To show these we introduce the space of hyperfunctions of growth which generalizes the space of distributions of growth and represent generalized functions as the initial values of smooth solutions of the heat equation.

     

Kuttner's problem and a Pólya type criterion for characteristic functions

Let be a continuous function with and . If is convex, then , , is the characteristic function of an absolutely continuous probability distribution. The criterion complements Pólya's theorem and applies to characteristic functions with various types of behavior at the origin. In particular, it provides upper bounds on Kuttner's function , , which gives the minimal value of such that is a characteristic function. Specifically, . Furthermore, improved lower bounds on Kuttner's function are obtained from an inequality due to Boas and Kac.

     

On the Gelfand-Kirillov conjecture for quantum algebras

Let be a complex not a root of unity and be a semi-simple Lie -algebra. Let be the quantized enveloping algebra of Drinfeld and Jimbo, be its triangular decomposition, and the associated quantum group. We describe explicitly and as a quantum Weyl field. We use for this a quantum analogue of the Taylor lemma.

     

A Liouville-type theorem on halfspaces for the Kohn Laplacian

Let be the Kohn Laplacian on the Heisenberg group and let be a halfspace of whose boundary is parallel to the center of . In this paper we prove that if is a non-negative -superharmonic function such that

then in .

     

Semi-classical limit of wave functions

We study in one dimension the semi-classical limit of the exact eigenfunction of the Hamiltonian , for a potential being analytic, bounded below and . The main result of this paper is that, for any given with two turning points, the exact normalized eigenfunction converges to the classical probability density, and the momentum distribution converges to the classical momentum density in the sense of distribution, as and with fixed. In this paper we only consider the harmonic oscillator Hamiltonian. By studying the semi-classical limit of the Wigner's quasi-probability density and using the generating function of the Laguerre polynomials, we give a complete mathematical proof of the Correspondence Principle.

     

On a Sobolev inequality with remainder terms

In this note we consider the Sobolev inequality

where is the best Sobolev constant and is the space obtained by taking the completion of with the norm . We prove here a refined version of this inequality,

where is a positive constant, the distance is taken in the Sobolev space , and is the set of solutions which attain the Sobolev equality. This generalizes a result of Bianchi and Egnell (A note on the Sobolev inequality, J. Funct. Anal. 100 (1991), 18-24), which was posed by Brezis and Lieb (Sobolev inequalities with remainder terms, J. Funct. Anal. 62 (1985), 73-86). regarding the classical Sobolev inequality

A key ingredient in our proof is the analysis of eigenvalues of the fourth order equation

where and is the unique radial function in with . We will show that the eigenvalues of the above equation are discrete:

and the corresponding eigenfunction spaces are

     

An uncertainty inequality involving norms

We derive a sharp uncertainty inequality of the form

with . As a consequence of this inequality we derive an upper bound for the so-called Laue constant, that is, the infimum of the functional , taken over all with (). Precisely, we obtain that which improves a previous bound of T. Gneiting.

     

Rearrangement of Hardy-Littlewood maximal functions in Lorentz spaces

For the classical Hardy-Littlewood maximal function , a well known and important estimate due to Herz and Stein gives the equivalence . In the present note, we study the validity of analogous estimates for maximal operators of the form

where denotes the Lorentz space -norm.

     

A finiteness theorem for a class of exponential congruences

For given elements and belonging to the ring of integers of a number field we consider the set of all tuples in for which divides for any and prove under some reasonable assumptions that the set of solutions is finite.

     

The Dynkin system generated by balls in contains all Borel sets

We show that for every each Borel subset of the space with the Euclidean metric can be generated from closed balls by complements and countable disjoint unions.