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 .

     

Tensor products of subnormal operators

We shall use a -algebra approach to study operators of the form where is subnormal and is normal. We shall determine the spectral properties for these operators, and find the minimal normal extension and the dual operator. We also give a necessary condition for to contain a compact operator and a sufficient condition for the algebraic equivalence of and .

We also consider the existence of a homomorphism satisfying . We shall characterize the operators such that exists for every operator .

The problem of when is unitarily equivalent to is considered. Complete results are given when and are positive operators with finite multiplicity functions and has compact self-commutator. Some examples are also given.

     

Opial's modulus and fixed points of semigroups of mappings

If is a Banach space with the non-strict Opial property and and is a nonempty convex weakly compact subset of , then every semigroup of asymptotically regular selfmappings of with has a common fixed point.

     

A lower bound for the number of components of the moduli schemes of stable rank 2 vector bundles on projective 3-folds

Fix a smooth projective 3-fold , , with ample, and . Assume the existence of integers with such that is numerically equivalent to . Let be the moduli scheme of -stable rank 2 vector bundles, , on with and . Let be the number ofits irreducible components. Then .

     

A non-metrizable compact linearly ordered topological space, every subspace of which has a -minimal base

A collection of subsets of a space is minimal if each element of contains a point which is not contained in any other element of . A base of a topological space is -minimal if it can be written as a union of countably many minimal collections. We will construct a compact linearly ordered space satisfying that is not metrizable and every subspace of has a -minimal base for its relative topology. This answers a problem of Bennett and Lutzer in the negative.

     

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 the existence of maximal Cohen-Macaulay modules over root extensions

Let be an unramified regular local ring having mixed characteristic and the integral closure of in a th root extension of its quotient field. We show that admits a finite, birational module such that . In other words, admits a maximal Cohen-Macaulay module.

     

Ample divisors on the blow up of at points

Fix integers with and ; if assume . Let be general points of the complex projective space and let be the blow up of at with exceptional divisors , . Set . Here we prove that the divisor is ample if and only if , i.e. if and only if .

     

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 groups with commutators of bounded order

Let be a prime, a non-negative integer. We prove that if is a residually finite group such that for all , then is locally finite.

     

Extremal points of a functional on the set of convex functions

We investigate the extremal points of a functional , for a convex or concave function . The admissible functions are convex themselves and satisfy a condition . We show that the extremal points are exactly and if these functions are convex and coincide on the boundary . No explicit regularity condition is imposed on , , or . Subsequently we discuss a number of extensions, such as the case when or are non-convex or do not coincide on the boundary, when the function also depends on , etc.

     

On a theorem of Scott and Swarup

Let be an exact sequence of hyperbolic groups induced by an automorphism of the free group . Let be a finitely generated distorted subgroup of . Then there exist and a free factor of such that the conjugacy class of is preserved by and contains a finite index subgroup of a conjugate of . This is an analog of a theorem of Scott and Swarup for surfaces in hyperbolic 3-manifolds.

     

Polynomially convex hulls of graphs on the sphere

Let be the graph of a continuous map of the unit sphere of into , and the polynomially convex hull of . Several examples of for are given, which have different properties from the known ones for .

     

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 .

     

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.

     

On the semisimplicity of pure sheaves

We obtain a criteria for a pure sheaf to be semisimple. As a corollary, we prove the following: Let and be two schemes over a finite field , and let be a proper smooth morphism. Assume is normal and geometrically connected, and assume there exists a closed point in such that the Frobenius automorphism acts semisimply on , where is the geometric fiber of at (this last assumption is unnecessary if the semisimplicity conjecture is true). Then is a semisimple sheaf on . This verifies a conjecture of Grothendieck and Serre provided the semisimplicity conjecture holds. As an application, we prove that the galois representations of function fields associated to the -adic cohomologies of surfaces are semisimple. We also get a theorem of Zarhin about the semisimplicity of the Galois representations of function fields arising from abelian varieties. The proof relies heavily on Deligne's work on Weil conjectures.

     

Semilinear transformations

In previous papers, nice trinomial equations were given for unramified coverings of the once punctured affine line in nonzero characteristic with the projective general group and the general linear group as Galois groups where is any integer and is any power of . These Galois groups were calculated over an algebraically closed ground field. Here we show that, when calculated over the prime field, as Galois groups we get the projective general semilinear group and the general semilinear group . We also obtain the semilinear versions of the local coverings considered in previous papers.

     

A class of 3-dimensional manifolds with bounded first eigenvalue on 1-forms

Let be the framebundle over an oriented, Riemannian surface . Denote by the first nonzero eigenvalue of the Laplace operator acting on differential forms of degree 1. We prove that for all with canonical metrics of volume 1.

     

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.

     

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 .