-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.

     

Sur les -corps

Here we give new examples of fields in characteristic whose -invariant and -invariant are different: or . These fields are also -fields.

RSUM. Nous donnons ici de nouveaux exemples de corps en caractéristique dont le -invariant et le -invariant diffèrent. Plus précisément: et ou . Ces corps sont aussi des -corps.

     

On the dimension of almost -dimensional spaces

Oversteegen and Tymchatyn proved that homeomorphism groups of positive dimensional Menger compacta are -dimensional by proving that almost -dimensional spaces are at most -dimensional. These homeomorphism groups are almost -dimensional and at least -dimensional by classical results of Brechner and Bestvina. In this note we prove that almost -dimensional spaces for are -dimensional. As a corollary we answer in the affirmative an old question of R. Duda by proving that every hereditarily locally connected, non-degenerate, separable, metric space is -dimensional.

     

Nonexpansive, -continuous antirepresentations have common fixed points

Let be a closed convex subset of a Banach (dual Banach) space . By we denote an antirepresentation of a semitopological semigroup as nonexpansive mappings on . Suppose that the mapping is jointly continuous when has the weak (weak*) topology and the Banach space of bounded right uniformly continuous functions on has a right invariant mean. If is weakly compact (for some the set is weakly* compact) and norm separable, then has a common fixed point in .

     

Sharper changes in topologies

Let be a Polish group, a Polish topology on a space , acting continuously on , with -invariant and in the Borel algebra generated by . Then there is a larger Polish topology on so that is open with respect to , still acts continuously on , and has a basis consisting of sets that are of the same Borel rank as relative to .

     

-identities on associative algebras

Let be an algebra over a field and a finite group of automorphisms and anti-automorphisms of . We prove that if satisfies an essential -polynomial identity of degree , then the -codimensions of are exponentially bounded and satisfies a polynomial identity whose degree is bounded by an explicit function of . As a consequence we show that if is an algebra with involution satisfying a -polynomial identity of degree , then the -codimensions of are exponentially bounded; this gives a new proof of a theorem of Amitsur stating that in this case must satisfy a polynomial identity and we can now give an upper bound on the degree of this identity.

     

type free resolutions of monomial ideals

Let be a monomial ideal of . Bayer-Peeva-Sturmfels studied a subcomplex of the Taylor resolution, defined by a simplicial complex . They proved that if is generic (i.e., no variable appears with the same non-zero exponent in two distinct monomials which are minimal generators), then is the minimal free resolution of , where is the Scarf complex of . In this paper, we prove the following: for a generic (in the above sense) monomial ideal and each integer , there is an embedded prime of . Thus a generic monomial ideal with no embedded primes is Cohen-Macaulay (in this case, is shellable). We also study a non-generic monomial ideal whose minimal free resolution is for some . In particular, we prove that if all associated primes of have the same height, then is Cohen-Macaulay and is pure and strongly connected.

     

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.

     

On the range and the kernel of the operator

Let denote the algebra of (bounded linear) operators on the separable complex Hilbert space , and let denote a norm ideal in . For , let the derivation be defined by , and let be defined by . The main result of this paper is to show that if , are contractions, then for every operator such that , then for all .

     

New facts

We use ``iterated square sequences' to show that there is an -definable partition such that if is an inner model not containing :

(a)

For some is stationary.

(b)

For each there is a generic extension of in which does not exist and is non-stationary.

This result is then applied to show that if is an inner model without , then some sentence not true in can be forced over .

     

Some corollaries of Frobenius' normal -complement theorem

For a prime divisor of the order of a finite group , we present the set of -subgroups generating . In particular, we present the set of primary subgroups of generating the last member of the lower central series of . The proof is based on the Frobenius Normal -Complement Theorem and basic properties of minimal nonnilpotent groups. Let be a group and a group-theoretic property inherited by subgroups and epimorphic images such that all minimal non--subgroups (-subgroups) of are not nilpotent. Then (see the lemma), if is generated by all -subgroups of it follows that is a -group.

     

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

     

Projections from

Let and be two infinite dimensional real Banach spaces. The following question is classical and long-standing. Are the following properties equivalent?

a) There exists a projection from the space of continuous linear operators onto the space of compact linear operators.

b) .

The answer is positive in certain cases, in particular if or has an unconditional basis. It seems that there are few results in the direction of a general solution. For example, suppose that and are reflexive and that or has the approximation property. Then, if , there is no projection of norm 1, from onto . In this paper, one obtains, in particular, the following result:

Theorem. Let be a real Banach space which is reflexive (resp. with a separable dual), of infinite dimension, and such that has the approximation property. Let be a real scalar with . Then can be equivalently renormed such that, for any projection from onto , one has . One gives also various results with two spaces and .

     

On sums and products of integers

Erdös and Szemerédi proved that if is a set of positive integers, then there must be at least integers that can be written as the sum or product of two elements of , where is a constant and . Nathanson proved that the result holds for . In this paper it is proved that the result holds for and .

     

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 .

     

Harmonic maps with noncontact boundary values

Every rank one symmetric space , of noncompact type, admits a compactification by attaching a sphere at infinity. If does not have constant sectional curvature, then admits a natural contact structure. This paper presents a number of harmonic maps , from to , which extend continuously to , and have noncontact boundary values. If the boundary values are assumed continuously differentiable, then the contact structure must be preserved.

     

An application of the regularized Siegel-Weil formula on unitary groups to a theta lifting problem

Let and be the pair of unitary groups over a global field and an irreducible cuspidal representation of which satisfies a certain -function condition. By using a regularized Siegel-Weil formula, we can show that the global theta lifting of in is non-trivial if every local factor of has a local theta lifting (Howe lifting) in .

     

Lusin's theorem for derivatives with respect to a continuous function

For a nowhere constant continuous function on a real interval and for a Borel measure on , we give simple necessary and sufficient conditions guaranteeing, for any Borel function on , the existence of a continuous function on such that the derivative of with respect to is, almost everywhere, equal to .

     

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 .

     

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 .