Existence of Bade functionals for complete Boolean algebras of projections in Fréchet spaces

A classical result of W. Bade states that if is any complete Boolean algebra of projections in an arbitrary Banach space then, for every there exists an element (called a Bade functional for with respect to in the dual space , with the following two properties: (i) is non-negative on and, (ii) whenever satisfies It is shown that a Fréchet space has this property if and only if it does not contain an isomorphic copy of the sequence space

     

Semi-free actions of zero-dimensional compact groups on Menger compacta

Let be the -dimensional universal Menger compactum, a -set in and a metrizable zero-dimensional compact group with the unit. It is proved that there exists a semi-free -action on such that is the fixed point set of every . As a corollary, it follows that each compactum with can be embedded in as the fixed point set of some semi-free -action on .

     

-bundles over aspherical surfaces and 4-dimensional geometries

Melvin has shown that closed 4-manifolds that arise as -bundles over closed, connected aspherical surfaces are classified up to diffeomorphism by the Stiefel-Whitney classes of the associated bundles. We show that each such 4-manifold admits one of the geometries or [depending on whether or ]. Conversely a geometric closed, connected 4-manifold of type or is the total space of an -bundle over a closed, connected aspherical surface precisely when its fundamental group is torsion free. Furthermore the total spaces of -bundles over closed, connected aspherical surfaces are all geometric. Conversely a geometric closed, connected 4-manifold is the total space of an -bundle if and only if where is torsion free.

     

Commensurators of parabolic subgroups of Coxeter groups

Let be a Coxeter system, and let be a subset of . The subgroup of generated by is denoted by and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of in is the subgroup of in such that has finite index in both and . The subgroup can be decomposed in the form where is finite and all the irreducible components of are infinite. Let be the set of in such that for all . We prove that the commensurator of is . In particular, the commensurator of a parabolic subgroup is a parabolic subgroup, and is its own commensurator if and only if .

     

Cellular filtration of K-theory and determinants of -algebras

In this note, we will disprove the following conjecture raised by Exel-Loring: Let be a -algebra with trace and let be a determinant associated to . If is a continuous family of homomorphisms and is the canonical matrix valued function on which represents the Bott element in , then . It should be noticed that the conjecture has been proved by Exel-Loring for the case that is a smooth family of homomorphisms.

     

Central units of the integral group ring

There are very few cases known of nonabelian groups where the group of central units of , denoted , is nontrivial and where the structure of , including a complete set of generators, has been determined. In this note, we show that the central units of augmentation 1 in the integral group ring form an infinite cyclic group , and we explicitly find the generator .

     

Gröbner duality and multiple points in linearly general position

It is proved for each , , that a primary -dimensional scheme in of degree in linearly general position lies in a rational normal scroll of dimension .

     

Local automorphisms and derivations on

Let be an algebra. A mapping is called a -local automorphism if for every there is an automorphism , depending on and , such that and (no linearity, surjectivity or continuity of is assumed). Let be an infinite-dimensional separable Hilbert space, and let be the algebra of all linear bounded operators on . Then every -local automorphism is an automorphism. An analogous result is obtained for derivations.

     

The infinitesimal cone of a totally positive semigroup

Given a complex reductive linear algebraic group split over with a fixed pinning, it is shown that all elements of the Lie algebra infinitesimal to the totally positive subsemigroup of lie in the totally positive cone .

     

Note on faithful representations and a local property of Lie groups

Let be any analytic group, let be a maximal toroid of the radical of , and let be a maximal semisimple analytic subgroup of . If is the Lie algebra of , is the radical of , and is the center of , we show that has a faithful representation if and only if (i) , and (ii) has a faithful representation.

     

Collapsing successors of singulars

Let be a singular cardinal in , and let be a model such that for some -cardinal with . We apply Shelah's pcf theory to study this situation, and prove the following results. 1) is not a -c.c generic extension of . 2) There is no ``good scale for ' in , so in particular weak forms of square must fail at . 3) If then and also . 4) If then .

     

A Strict Version of the Non-commutative Urysohn Lemma

Given a pair , of -commuting, hereditary -subalgebras of a unital -algebra , such that is -unital and , there is an element in , with , such that is strictly positive in and is strictly positive in in . Moreover, is strictly positive in in .

     

On Bernstein type theorems concerning the growth of derivatives of entire functions

A subspace of which is invariant under all left translation operators is called admissible if is a Banach space satisfying the following properties:

(i) If then there exists a subsequence such that almost everywhere.

(ii) The group is a bounded strongly continuous group. In this case, let

Typical admissible spaces are and all spaces for More generally, all of the Peetre interpolation spaces of two admissible spaces are also admissible.

A function is called subexponential if for every With these definitions our main result goes as follows: . If is an entire function of exponential type such that its restriction to the real axis, denoted by , is subexponential and belongs to some admissible space then the derivative is also in Moreover,
for each real

This result yields as consequences and in a systematic way many new and old Bernstein type inequalities.

     

Combinatorial aspects of F-sets

We discuss F filters and show that the minimum size of a filter base generating an undiagonalizable filter included in some F filter is the better known bounded evasion number . An application to -sets from trigonometric series is given by showing that if is an -set and has size less than , then is again an -set.

     

Cohomological dimension and approximate limits

Approximate (inverse) systems of compacta have been useful in the study of covering dimension, dim, and cohomological dimension over an abelian group , . Such systems are more general than (classical) inverse systems. They have limits and structurally have similar properties. In particular, the limit of an approximate system of compacta satisfies the important property of being an approximate resolution. We shall prove herein that if is an abelian group, a compactum is the limit of an approximate system of compacta , , and for each , then .

     

A convolution estimate for a measure on a curve in

Let and fix an interval . If is the operator on defined by , then maps into .

     

The structure of hypersurfaces with some curvature conditions

Let be a hypersurface in , and let denote the mean curvature and the scalar curvature of respectively. We show that if is compact and , then is diffeomorphic to . Also we prove that if is complete, is constant and , then is or or .

     

A sequential property of and a covering property of Hurewicz

has the monotonic sequence selection property if there is for each , and for every sequence where for each is a sequence converging pointwise monotonically to , a sequence such that for each is a term of , and converges pointwise to . We prove a theorem which implies for metric spaces that has the monotonic sequence selection property if, and only if, has a covering property of Hurewicz.

     

Every local ring is dominated by a one-dimensional local ring

Let be a local (Noetherian) ring. The main result of this paper asserts the existence of a local extension ring of such that (i) dominates , (ii) the residue field of is a finite purely transcendental extension of , (iii) every associated prime of (0) in contracts in to an associated prime of (0), and (iv) . In addition, it is shown that can be obtained so that either is the maximal ideal of or is a localization of a finitely generated -algebra.

     

Beals-Cordes-type characterizations of pseudodifferential operators

We show that, if is the representation of on given by (2.11), and is a bounded operator on , then belongs to if and only if

is a function on with values in the Banach space .