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.

     

Eigenvalues of the form valued Laplacian for Riemannian submersions

Let be a Riemannian submersion of closed manifolds. Let be an eigen -form of the Laplacian on with eigenvalue which pulls back to an eigen -form of the Laplacian on with eigenvalue . We are interested in when the eigenvalue can change. We show that , so the eigenvalue can only increase; and we give some examples where , so the eigenvalue changes. If the horizontal distribution is integrable and if is simply connected, then , so the eigenvalue does not change.

     

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 .

     

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 Schatten space -algebra

For any , let denote the classical -Schatten space of operators on the Hilbert space . It was shown by Varopoulos (for ) and by Blecher and the author (full result) that for any equipped with the Schur product is an operator algebra. Here we prove that (and thus for any ) is actually a -algebra, which means that it is isomorphic to some quotient of a uniform algebra in the Banach algebra sense.

     

On certain character sums over

Let be the finite field with elements and let denote the ring of polynomials in one variable with coefficients in . Let be a monic polynomial irreducible in . We obtain a bound for the least degree of a monic polynomial irreducible in ( odd) which is a quadratic non-residue modulo . We also find a bound for the least degree of a monic polynomial irreducible in which is a primitive root modulo .

     

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 .

     

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 .

     

Unbounded gaps for cocycles and invariant measures for their Mackey actions

We show that for a class of type -cocycles over a -action of type its Mackey action must change the type.

     

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 .

     

Degrees of high-dimensional subvarieties of determinantal varieties

Let , where is a prime, and . In , let be the variety defined by . We show that any subvariety of of codimension less than must have degree a multiple of . We also show that the bounds on the codimension in our results are strict by exhibiting subvarieties of the appropriate codimension whose degrees are prime to .

     

A counterexample to a -analogue of the chromatic splitting conjecture

We prove that, if , the -localization of the -localization map is not a split monomorphism in the stable category by exhibiting spectra for which the map is not injective. If and , we show that may be taken to be a two-cell complex in the sense of -local homotopy theory. The question of whether the map splits was asked by Hovey and is in some sense a -analogue of Hopkins' chromatic splitting conjecture.

     

Probability measures in -algebras in Hilbert spaces with conjugation

Let be a real -algebra of -real bounded operators containing no central summand of type in a complex Hilbert space with conjugation . Denote by the quantum logic of all -orthogonal projections in the von Neumann algebra . Let be a probability measure. It is shown that contains a finite central summand and there exists a normal finite trace on such that , .

     

Properties of subgenerators of regularized semigroups

We introduce two operations , in the set of subgenerators of a given - regularized semigroup and prove that is a complete partially ordered lattice with respect to , and the operator inclusion . Also presented are some other properties and examples for

     

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.

     

A decreasing operator function associated with the Furuta inequality

Let with and let and . As a generalization of a result due to Furuta, it is shown that the operator function

is decreasing for and if . Moreover, if and , then is decreasing for and . The latter result is an extension of an earlier result of Furuta.

     

Common operator properties of the linear operators

Let and be bounded linear operators defined on Banach spaces, , . When , then the operators and have many basic operator properties in common. This situation is studied in this paper.

     

Ditkin's condition for certain Beurling algebras

Let be a locally compact abelian group. A function is said to be a weight if it is locally bounded, Borel measurable and submultiplicative. We call a weight on semi-bounded if there exist a constant and a subsemigroup with such that

for all Using functional analytic methods, we show that all Beurling algebras whose defining weight is semi-bounded satisfy Ditkin's condition.

     

Elementary abelian 2-group actions on flag manifolds and applications

Let denote the unoriented cobordism ring. Let and let denote the equivariant cobordism ring of smooth manifolds with smooth -actions having finite stationary points. In this paper we show that the unoriented cobordism class of the (real) flag manifold is in the subalgebra generated by , where , and . We obtain sufficient conditions for indecomposability of an element in . We also obtain a sufficient condition for algebraic independence of any set of elements in . Using our criteria, we construct many indecomposable elements in the kernel of the forgetful map in dimensions , for , and show that they generate a polynomial subalgebra of .