Multiplicative bijections of

Let be a compact Hausdorff space which satisfies the first axiom of countability, let and let , be the set of all continuous functions from to If , ,is a bijective multiplicative map, then there exist a homeomorphism and a continuous map such that for all and for all

     

Hyperbolic unit groups

In this paper we study the groups whose integral group rings have hyperbolic unit groups . We classify completely the torsion subgroups of and the polycyclic-by-finite subgroups of the group . Finally, we classify the groups for which the boundary of has dimension zero.

     

Sobolev spaces and the Cayley transform

The generalised Cayley transform  from an Iwasawa -group into the corresponding real unit sphere  induces isomorphisms between suitable Sobolev spaces and . We study the differential of  , and we obtain a criterion for a function to be in  .

     

Gradient ranges of bumps on the plane

For a -smooth bump function we show that the gradient range is the closure of its interior, provided that admits a modulus of continuity satisfying as . The result is a consequence of a more general result about gradient ranges of bump functions of the same degree of smoothness. For such bump functions we show that for open sets , either the intersection is empty or its topological dimension is at least two. The proof relies on a new Morse-Sard type result where the smoothness hypothesis is independent of the dimension of the space.

     

Hardy's inequality on exterior domains

Let be a smooth exterior domain in and . We prove that when , Hardy's inequality is valid on .

     

``Lebesgue measure' on , II

Let be the set of real numbers, and define . We construct a complete measure space where the -algebra contains the Borel subsets of , and is a translation-invariant measure such that for any measurable rectangle , if , then , where is Lebesgue measure on . The measure is not -finite. We prove three Fubini theorems, namely, the Fubini theorem, the mean Fubini-Jensen theorem, and the pointwise Fubini-Jensen theorem. Finally, as an application of the measure , we construct, via selfadjoint operators on , a ``Schrödinger model' of the canonical commutation relations: , , .

     

On the Betti numbers of sign conditions

Let be a real closed field and let and be finite subsets of such that the set has elements, the algebraic set defined by has dimension and the elements of and have degree at most . For each we denote the sum of the -th Betti numbers over the realizations of all sign conditions of on by . We prove that

This generalizes to all the higher Betti numbers the bound on . We also prove, using similar methods, that the sum of the Betti numbers of the intersection of with a closed semi-algebraic set, defined by a quantifier-free Boolean formula without negations with atoms of the form or for , is bounded by

making the bound more precise.

     

On the algebra of functions -extendable for each finite

For each positive integer we construct a -function of one real variable, the graph of which has the following property: there exists a real function on which is -extendable to , for each finite, but it is not -extendable.

     

-function of an operator: A white noise approach

Let be the canonical framework of white noise analysis over the Gel'fand triple and be the space of continuous linear operators from to . Let be a self-adjoint operator in with spectral representation . In this paper, it is proved that under appropriate conditions upon , there exists a unique linear mapping such that for each . The mapping is then naturally used to define as , where is the Dirac -function. Finally, properties of the mapping are investigated and several results are obtained.

     

On functions whose graph is a Hamel basis

We say that a function is a Hamel function ( ) if , considered as a subset of , is a Hamel basis for . We prove that every function from into can be represented as a pointwise sum of two Hamel functions. The latter is equivalent to the statement: for all there is a such that . We show that this fails for infinitely many functions.

     

Poincaré duality algebras and rings of coinvariants

Let be a faithful representation of a finite group over the field . Via the group acts on and hence on the algebra of homogenous polynomial functions on the vector space . R. Kane (1994) formulated the following result based on the work of R. Steinberg (1964): If the field has characteristic 0, then is a Poincaré duality algebra if and only if is a pseudoreflection group. The purpose of this note is to extend this result to the case (i.e. the order of is relatively prime to the characteristic of ).

     

Hilbert modules over a class of semicrossed products

Given the disk algebra and an automorphism , there is associated a non-self-adjoint operator algebra called the semicrossed product of with . Buske and Peters showed that there is a one-to-one correspondence between the contractive Hilbert modules over and pairs of contractions and on satisfying . In this paper, we show that the orthogonally projective and Shilov Hilbert modules over correspond to pairs of isometries on satisfying . The problem of commutant lifting for is left open, but some related results are presented.     

Cuntz-Pimnser algebras, completely positive maps and Morita equivalence

Let be a completely positive map on and let be the associated GNS--correspondence. We prove a result that implies, in particular, that the Cuntz-Pimsner algebra of , , is strongly Morita equivalent to the Cuntz algebra , where is the index of .

     

A multiplicity theorem for the Neumann problem

Here is a particular case of the main result of this paper: Let be a bounded domain, with a boundary of class , and let be two continuous functions, , with 0$">, , with n$">. If

and if the set of all global minima of the function has at least connected components, then, for each 0$"> small enough, the Neumann problem

admits at least strong solutions in .

     

Tensor products of -weakly closed nest algebra submodules

In this paper we prove that for any unital -weakly closed algebra which is -weakly generated by finite-rank operators in , every -weakly closed -submodule has . In the case of nest algebras, if are nests, we obtain the following -fold tensor product formula:

where each is the -weakly closed Alg -submodule determined by an order homomorphism from into itself.

     

Segal-Bargmann transforms of one-mode interacting Fock spaces associated with Gaussian and Poisson measures

Let and denote the Gaussian and Poisson measures on , respectively. We show that there exists a unique measure on such that under the Segal-Bargmann transform the space is isomorphic to the space of analytic -functions on with respect to . We also introduce the Segal-Bargmann transform for the Poisson measure and prove the corresponding result. As a consequence, when and have the same variance, and are isomorphic to the same space under the - and -transforms, respectively. However, we show that the multiplication operators by on and on act quite differently on .

     

Linear functionals on the Cuntz algebra

For a pure state on , which is an extension of a pure state on with the property that if is a corresponding representation, then , induces a unital shift of of the Powers index . We describe states on by using sequences of unit vectors in . We study the linear functionals on the Cuntz algebra whose restrictions are the product pure state on . We find conditions on the sequence of unit vectors for which the corresponding linear functionals on become states under these conditions.

     

Minimal sufficiency of order statistics in convex models

Let be a convex and dominated statistical model on the measurable space , with minimal sufficient, and let . Then , the -algebra of all permutation invariant sets belonging to the -fold product -algebra , is shown to be minimal sufficient for the corresponding model for independent observations, .

The main technical tool provided and used is a functional analogue of a theorem of Grzegorek (1982) concerning generators of .

     

-bounded groups and other topological groups with strong combinatorial properties

We construct several topological groups with very strong combinatorial properties. In particular, we give simple examples of subgroups of (thus strictly -bounded) which have the Menger and Hurewicz properties but are not -compact, and show that the product of two -bounded subgroups of may fail to be -bounded, even when they satisfy the stronger property . This solves a problem of Tkacenko and Hernandez, and extends independent solutions of Krawczyk and Michalewski and of Banakh, Nickolas, and Sanchis. We also construct separable metrizable groups of size continuum such that every countable Borel -cover of contains a -cover of .

     

A perturbed elementary operator and range-kernel orthogonality

Let denote the algebra of operators on a Hilbert . If and are commuting normal operators, and and are commuting quasi-nilpotents such that , then define and by , , and . It is proved that and , where is some scalar and is the quasi-nilpotent part of the operator .