Zero divisors and

Let be a discrete group, the group ring of over and the Lebesgue space of with respect to Haar measure. It is known that if is torsion free elementary amenable, and , then . We will give a sufficient condition for this to be true when , and in the case we will give sufficient conditions for this to be false when .

     

Examples of chain domains

Let be a nonzero ordinal such that for every ordinal . A chain domain (i.e. a domain with linearly ordered lattices of left ideals and right ideals) is constructed such that is isomorphic with all its nonzero factor-rings and is the ordinal type of the set of proper ideals of . The construction provides answers to some open questions.

     

A classification of all is a Prüfer domain

Let be an integral domain with quotient field . The ring of integer-valued polynomials over is defined by . It is known that if is a Prüfer domain, then is an almost Dedekind domain with all residue fields finite. This condition is necessary and sufficient if is Noetherian, but has been shown to not be sufficient if is not Noetherian. Several authors have come close to a complete characterization by imposing bounds on orders of residue fields of and on normalized values of particular elements of . In this note we give a double-boundedness condition which provides a complete charaterization of all integral domains such that is a Prüfer domain.

     

A generalization of Lyapunov's convexity theorem with applications in optimal stopping

Lyapunov proved that the range of finite measures defined on the same -algebra is compact, and if each measure also is atomless, then the range is convex. Although both conclusions may fail for measures on different -algebras of the same set, they do hold if the -algebras are nested, which is exactly the setting of classical optimal stopping theory.

     

Stability of the surjectivity of endomorphisms and isometries of

We determine the largest positive number with the property that whenever are endomorphisms, respectively unital isometries of the algebra of all bounded linear operators acting on a separable Hilbert space, holds for every nonzero and is surjective, then so is . It turns out that in the first case we have , while in the second one .

     

On uniqueness of invariant means

The following results on uniqueness of invariant means are shown:

(i) Let be a connected almost simple algebraic group defined over . Assume that , the group of the real points in , is not compact. Let be a prime, and let be the compact -adic Lie group of the -points in . Then the normalized Haar measure on is the unique invariant mean on .

(ii) Let be a semisimple Lie group with finite centre and without compact factors, and let be a lattice in . Then integration against the -invariant probability measure on the homogeneous space is the unique -invariant mean on .

     

-convexity of Orlicz-Bochner spaces

A characterization of -convexity of arbitrary Banach space is given. Moreover, it is proved that the Orlicz-Bochner function space is P-convex if and only if both spaces and are -convex. In particular, the Lebesgue-Bochner space with is -convex iff is -convex.

     

Optimal estimation of shell thickness in Cutland's construction of Wiener measure

In Cutland's construction of Wiener measure, he used the product of Gaussian measures on , where is an infinite integer. It is mentioned by Cutland and Ng that for the product measure ,

where and with any positive infinite number. We prove here that may be replaced by with any positive infinite number. This is the optimal estimation for the shell thickness. It is also proved that . And for the *Lebesgue measure , is finite and not infinitesimal iff with finite, while for the *Lebesgue area of the sphere , should be .

     

Continuity of K-theory: An example in equal characteristics

If is a perfect field of characteristic , we show that the Quillen K-groups are uniquely -divisible for . In fact, the Milnor K-groups are uniquely -divisible for all . This implies that is -connected after profinite completion for a complete discrete valuation ring with perfect residue field.

     

A counterexample to a question of R. Haydon, E. Odell and H. Rosenthal

We give an example of a compact metric space , an open dense subset of , and a sequence in which is pointwise convergent to a non-continuous function on , such that for every there exists with for all , yet is equivalent to the unit vector basis of the James quasi-reflexive space of order 1. Thus does not embed isomorphically in the closed linear span of . This answers in the negative a question asked by H. Haydon, E. Odell and H. Rosenthal.

     

Average curvature of convex curves in

A well-known result states that, if a curve in has geodesic curvature less than or equal to one at every point, then is embedded. The converse is obviously not true, but the embeddedness of a curve does give information about the curvature. We prove that, if is a convex embedded curve in , then the average curvature (curvature per unit length) of , denoted , satisfies . This bound on the average curvature is tight as for a horocycle.

     

The 6-property for simplicial complexes and a combinatorial Cartan-Hadamard theorem for manifolds

The 6-property for 2-dimensional simplicial complexes is the condition that every nontrivial circuit in the link of a vertex has length greater than or equal to six. If a compact -manifold has a 2-dimensional spine with the 6-property, then we show that the interior of is covered by euclidean -space. In dimension , we show further that such a 3-manifold is Haken.

     

On an Isomorphism Problem for Endomorphism Near-Rings

Suppose and are endomorphism near-rings generated by
groups of automorphisms containing the inner automorphisms of two respective finite perfect groups and . In this note we show that if and are isomorphic, then and are isomorphic.

     

Extendibility of homogeneous polynomials on Banach spaces

We study the -homogeneous polynomials on a Banach space that can be extended to any space containing . We show that there is an upper bound on the norm of the extension. We construct a predual for the space of all extendible -homogeneous polynomials on and we characterize the extendible 2-homogeneous polynomials on when is a Hilbert space, an -space or an -space.

     

The moduli of substructures of a compact complex space

We construct a space of fine moduli for the substructures of an arbitrary compact complex space . A substructure of is given by a subalgebra of the structure sheaf with the additional feature that is also a complex space; and are called equivalent if and only if and are isomorphic as subalgebras of .

Since substructures are quotients, it is only natural to start with the fine moduli space of all complex-analytic quotients of . In order to obtain a representable moduli functor of substructures, we are forced to concentrate on families of quotients which satisfy some flatness condition for relative differential modules of higher order. Considering the corresponding flatification of , we realize that its open subset consisting of all substructures turns out to be a complex space which has the required universal property.

     

Some results on finite Drinfeld modules

Let be a global function field, a degree one prime divisor of and let be the Dedekind domain of functions in regular outside . Let be the Hilbert class field of , the integral closure of in . Let be a rank one normalized Drinfeld -module and let be a prime ideal in . We explicitly determine the finite -module structure of . In particular, if , is an odd prime number and is the Carlitz -module, then the finite -module is always cyclic.

     

The stability radius of a quasi-Fredholm operator

We extend the technique used by Kordula and Müller to show that the stability radius of a quasi-Fredholm operator is the limit of as . If is an isolated point of the Apostol spectrum , then the above limit is non-zero if and only if is quasi-Fredholm.

     

Duality and local group cohomology

Let be a group, let be a field, and let be a local system - an upwardly directed collection of subgroups whose union is . In this paper we give a short, elementary proof of the following result: If either is a --bimodule, or else is finite dimensional over its center, then . From this we deduce as easy corollaries some recent results of Meierfrankenfeld and Wehrfritz on the cohomology of a finitary module.

     

Metrizability of sequential topological groups with point-countable -networks

We prove that a Hausdorff sequential topological group with a point-countable -network is metrizable iff its sequential order is less than . In the non Hausdorff case metrizability may be replaced by -locally finite base.

     

Pseudo-uniform convergence, a nonstandard treatment

We introduce and study the notion of pseudo-uniform convergence which is a weaker variant of quasi-uniform convergence. Applications include the following nonstandard characterization of weak convergence. Let be an infinite set, the Banach space of all bounded real-valued functions on a bounded sequence in and Then the sequence converges weakly to if and only if the convergence is pointwise on and, for each strictly increasing function , each , and each , there is an unlimited such that .