An extension of H. Cartan's theorem

In this article we prove that if , , is a bounded pseudoconvex domain with real analytic boundary, then for each , there exists a fixed open neighborhood of and an open neighborhood of in such that any can be extended holomorphically to , and that the action defined by

is real analytic in joint variables. This extends H. Cartan's theorem beyond the boundary. Some applications are also discussed here.

     

On covering multiplicity

Let be a system of arithmetic sequences which forms an -cover of (i.e. every integer belongs at least to members of ). In this paper we show the following surprising properties of : (a) For each there exist at least subsets of with such that . (b) If forms a minimal -cover of , then for any there is an such that for every there exists an for which and

     

-compactness and automatic continuity in ultrametric spaces of bounded continuous functions

In this paper (weakly) separating maps between spaces of bounded continuous functions over a nonarchimedean field are studied. It is proven that the behaviour of these maps when is not locally compact is very different from the case of real- or complex-valued functions: in general, for -compact spaces and , the existence of a (weakly) separating additive map implies that and are homeomorphic, whereas when dealing with real-valued functions, this result is in general false, and we can just deduce the existence of a homeomorphism between the Stone-Cech compactifications of and . Finally, we also describe the general form of bijective weakly separating linear maps and deduce some automatic continuity results.

     

On the product of two generalized derivations

Two elements and in a ring determine a generalized derivation on by setting for any in . We characterize when the product is a generalized derivation in the cases when the ring is the algebra of all bounded operators on a Banach space , and when is a -algebra . We use these characterizations to compute the commutant of the range of .

     

On a problem of Dynkin

Consider an -superdiffusion on , where is an uniformly elliptic differential operator in , and . The -polar sets for are subsets of which have no intersection with the graph of , and they are related to the removable singularities for a corresponding nonlinear parabolic partial differential equation. Dynkin characterized the -polarity of a general analytic set in term of the Bessel capacity of , and Sheu in term of the restricted Hausdorff dimension. In this paper we study in particular the -polarity of sets of the form , where and are two Borel subsets of and respectively. We establish a relationship between the restricted Hausdorff dimension of and the usual Hausdorff dimensions of and . As an application, we obtain a criterion for -polarity of in terms of the Hausdorff dimensions of and , which also gives an answer to a problem proposed by Dynkin in the 1991 Wald Memorial Lectures.

     

The distribution of solutions of the congruence

For a cube of size , we obtain a lower bound on so that is nonempty, where is the algebraic subset of defined by

a positive integer and an integer not divisible by . For we obtain that is nonempty if , for we obtain that is nonempty if , and for we obtain that is nonempty if . Using the assumption of the Grand Riemann Hypothesis we obtain is nonempty if .

     

A structure of ring homomorphisms on commutative Banach algebras

We give a structure theorem for a ring homomorphism of a commutative regular Banach algebra into another commutative Banach algebra. In particular, it is shown that:

(i)

A ring homomorphism of a commutative -algebra onto another commutative -algebra with connected infinite Gelfand space is either linear or anti-linear.

(ii)

A ring automorphism of is either linear or anti-linear.

(iii)

, and the disc algebra are neither ring homomorphic images of nor .

     

Exact estimates for integrals involving Dirichlet series with nonnegative coefficients

We consider the Dirichlet series

with coefficients for all . Among others, we prove exact estimates of certain weighted -norms of on the unit interval for any , in terms of the coefficients . Our estimation is based on the close relationship between Dirichlet series and power series. This enables us to derive exact estimates for integrals involving the former one by relying on exact estimates for integrals involving the latter one. As a by-product, we obtain an analogue of the Cauchy-Hadamard criterion of (absolute) convergence of the more general Dirichlet series

with complex coefficients .

     

Complete positivity of elementary operators

In this paper, we prove that if is an -dimensional subspace of , then is -reflexive, where denotes the greatest integer not larger than . By the result, we show that if is an elementary operator on a -algebra , then is completely positive if and only if is -positive.

     

Normalizers of the congruence subgroups of the Hecke group

Let cos and let be the Hecke group associated to . In this article, we show that for a prime ideal in , the congruence subgroups of are self-normalized in .

     

Relative modular theory for a weight

We consider the balanced weight of a semi-finite weight and a (not necessarily faithful) normal positive functional on a von Neumann algebra , and discuss how the modular operator and the modular conjugation are described under the identification of the standard Hilbert space with , where is the support projection of and .

     

A factorization theorem for the derivative of a function in

We show that a function is the derivative of a function in the Hardy space of the unit disk for if and only if where and . Here, can be chosen to be non-vanishing, , and . As an application, we characterize positive measures on the unit disk such that the operator is bounded from the tent space to , where .

     

Irreducible plane curves with the Albanese dimension 2

Let be a plane curve given by an equation , and let be the affine plane curve given by . Let denote a cyclic covering of determined by . The number is called the Albanese dimension of . In this article, we shall give examples of with the Albanese dimension 2.

     

Fundamental theorem of geometry without the 1-to-1 assumption

It is proved that any mapping of an -dimensional affine space over a division ring onto itself which maps every line into a line is semi-affine, if and . This result seems to be new even for the real affine spaces. Some further generalizations are also given. The paper is self-contained, modulo some basic terms and elementary facts concerning linear spaces and also - if the reader is interested in other than , , or - division rings.

     

Brownian space-time functions of zero quadratic variation depend only on time

Let , , be a -dimensional Brownian motion and let be a continuous function. We show that if is locally of zero quadratic variation, then for all . This result extends recent work of F. B. Knight, thereby confirming a conjecture of T.Salisbury.

     

Global bifurcation in generic systems of nonlinear Sturm-Liouville problems

We consider the system of coupled nonlinear Sturm-Liouville boundary value problems where , are real spectral parameters. It will be shown that if the functions and are `generic' then for all integers , there are smooth 2-dimensional manifolds , , of `semi-trivial' solutions of the system which bifurcate from the eigenvalues , , of , , respectively. Furthermore, there are smooth curves , , along which secondary bifurcations take place, giving rise to smooth, 2-dimensional manifolds of `non-trivial' solutions. It is shown that there is a single such manifold, , which `links' the curves , . Nodal properties of solutions on and global properties of are also discussed.

     

Relations between the Taylor spectrum and the Xia spectrum

Let be a doubly commuting -tuple of -hyponormal operators with unitary operators from the polar decompositions . Let and . In this paper, we will show relations between the Taylor spectrum and the Xia spectrum .

     

Braided tensor C*-categories, Hecke symmetries and actions on extended Cuntz algebras

In this paper we deal with braided tensor C*-categories. For every object of the category we associate a C*-algebra denoted by . An analysis of the braiding is carried out by using the conjugate equations. If the braiding is a Hecke symmetry and the -dimension is appropriately chosen, we characterize the C*-algebra as the one generated by the representation given by the Markov trace. This analysis leads to the existence of an action of on . Such actions (Theorem 1) correspond to *-monomorphisms of on which generalize the ones obtained earlier by the author (1997).

     

Open covers and partition relations

An open cover of a topological space is said to be an -cover if there is for each finite subset of the space a member of the cover which contains the finite set, but the space itself is not a member of the cover. We prove theorems which imply that a set of real numbers has Rothberger's property if, and only if, for each positive integer , for each -cover of , and for each function from the two-element subsets of , there is a subset of such that is constant on , and each element of belongs to infinitely many elements of (Theorem 1). A similar characterization is given of Menger's property for sets of real numbers (Theorem 6).

     

The connected stable rank of the purely infinite simple -algebras

Suppose that is a unital purely infinite simple -algebra. If the class [1] of the unit 1 in has torsion, then ; if [1] is torsion-free in , then . If is a non-unital purely infinite simple -algebra, then .