Paracompact subspaces in the box product topology

In 1975 E. K. van Douwen showed that if is a family of Hausdorff spaces such that all finite subproducts are paracompact, then for each element of the box product the -product is paracompact. He asked whether this result remains true if one considers uncountable families of spaces. In this paper we prove in particular the following result: Let be an infinite cardinal number, and let be a family of compact Hausdorff spaces. Let be a fixed point. Given a family of open subsets of which covers , there exists an open locally finite in refinement of which covers .

We also prove a slightly weaker version of this theorem for Hausdorff spaces with ``all finite subproducts are paracompact" property. As a corollary we get an affirmative answer to van Douwen's question.

     

Bounds for the operator norms of some Nörlund matrices

Suppose is a non-increasing sequence of non-negative numbers with , , , and is the lower triangular matrix defined by , , and , . We show that the operator norm of as a linear operator on is no greater than , for ; this generalizes, yet again, Hardy's inequality for sequences, and simplifies and improves, in this special case, more generally applicable results of D. Borwein, Cass, and Kratz. When the tend to a positive limit, the operator norm of on is exactly . We also give some cases when the operator norm of on is less than .

     

Isometries of certain operator algebras

To a given basis on an -dimensional Hilbert space , we associate the algebra of all linear operators on having every as an eigenvector. So, is commutative, semisimple, and -dimensional. Given two algebras of this type, and , there is a natural algebraic isomorphism of and . We study the question: When does preserve the operator norm?

     

Parametrizing maximal compact subvarieties

For the Lie group , let be the open orbit of Lagrangian planes of signature in the generalized flag variety of Lagrangian planes in . For a suitably chosen maximal compact subgroup of and a base point we have that the orbit of is a maximal compact subvariety of . We show that for the connected component containing in the space of translates of which lie in is biholomorphic to , where denotes with the opposite complex structure.

     

Banach spaces in which every -weakly summable sequence lies in the range of a vector measure

Let be a Banach space. For we prove that the identity map is -summing if and only if the operator is nuclear for every unconditionally summable sequence in , where is the conjugate number for . Using this result we find a characterization of Banach spaces in which every -weakly summable sequence lies inside the range of an -valued measure (equivalently, every -weakly summable sequence in , satisfying that the operator is compact, lies in the range of an -valued measure) with bounded variation. They are those Banach spaces such that the identity operator is -summing.

     

A note on generators of least degree in Gorenstein ideals

Assume is a polynomial ring over a field and is a homogeneous Gorenstein ideal of codimension and initial degree . We prove that the number of minimal generators of that are of degree is bounded above by , which is the number of minimal generators of the defining ideal of the extremal Gorenstein algebra of codimension and initial degree . Further, is itself extremal if .

     

Almost linearity of -bi-Lipschitz maps between real Banach spaces

Let and be real Banach spaces. A map between and is called an -bi-Lipschitz map if for all . In this note we show that if is an -bi-Lipschitz map with from onto , then is almost linear. We also show that if is a surjective -bi-Lipschitz map with , then there exists a linear isomorphism such that

where as and .

     

Cohomological detection and regular elements in group cohomology

Suppose that is a compact Lie group or a discrete group of finite virtual cohomological dimension and that is a field of characteristic . Suppose that is a set of elementary abelian -subgroups such that the cohomology is detected on the centralizers of the elements of . Assume also that is closed under conjugation and that is in whenever some subgroup of is in . Then there exists a regular element in the cohomology ring such that the restriction of to an elementary abelian -subgroup is not nilpotent if and only if is in . The converse of the result is a theorem of Lannes, Schwartz and the second author. The results have several implications for the depth and associated primes of the cohomology rings.

     

Symplectic leaves and deformation quantization

In this paper, we show that for any classical simple compact Poisson Lie group , there is no quantization of using the quantum group , which is both group-preserving and symplectic leaf-preserving.

     

The local zeta function for the non-trivial characters associated with the singular Jordan algebras

This paper investigates the local integrals

where represents the integers of a composition algebra over a non-archimedean local field and is a non-trivial character on the units in the ring of integers of extended to by setting . The local zeta function for the trivial character is known for all composition algebras . In this paper, we show in the quaternion case that for all non-trivial characters and then compute the local zeta function in the ramified quadratic extension case for equal to the quadratic character. In this latter case, for any character of order greater than .

     

Lipscomb's universal space is the attractor of an infinite iterated function system

Lipscomb's one-dimensional space on an arbitrary index set is injected into the Tychonoff cube . The image of is shown to be the attractor of an iterated function system indexed by . This system is conjugate, under an injection, with a set of right-shift operators on Baire's space regarded as a code space. This view of extends the fractal nature of initiated in a 1992 joint paper by the author and S. Lipscomb. In addition, we give a new proof that as a subspace of Hilbert's space , the space is complete and hence is closed in .

     

The -PIP and integrability of a single function

Two examples are given that answer in the negative the following question asked by E. M. Bator: If is bounded and weakly measurable and for each in there is a bounded sequence in such that a.e., does it follow that is Pettis integrable?

     

Properties that characterize Gaussian periods and cyclotomic numbers

Let be a prime number, a -th primitive root of 1 and the periods of degree of . Write with . Several characterizations of the numbers and (or, equivalently, of the cyclotomic numbers of order ) are given in terms of systems of equations they satisfy and a condition on the linear independence, over , of the or on the irreducibility, over , of the characteristic polynomial of the matrix .

     

and the associated line bundles

Let be a compact semi-simple Lie group, and let be a maximal unipotent subgroup of the complexified group . In this paper, we classify all the -invariant Kaehler structures on . For each Kaehler structure , let be the line bundle with connection whose curvature is . We then study the holomorphic sections of , which constitute a -representation space.

     

On a question ofMakar-Limanov

Let be an uncountable field, let be a field extension, and let be an associative -algebra. We show that if contains a non-commutative free algebra, then so does .

     

Weighted inequalities for the maximal geometric mean operator

For nonnegative Borel measures on and for the maximal geometric mean operator , we characterize the weight pairs for which is of weak type and of strong type , . No doubling conditions are needed. We also note that a previously published different characterization for the strong type inequality for has an incorrect proof.

     

Proof of the Simon-Ando Theorem

In 1961, Simon and Ando wrote a classical paper describing the convergence properties of nearly completely decomposable matrices. Basically, their work concerned a partitioned stochastic matrix e.g.

where and are square blocks whose entries are all larger than those of and respectively.

Setting

partitioned as in , they observed that for some, rather short, initial sequence of iterates the main diagonal blocks tended to matrices all of whose rows are identical, e.g. to and to . After this initial sequence, subsequent iterations showed that all blocks lying in the same column as those matrices tended to a scalar multiple of them, e.g.

where and .

The purpose of this paper is to give a qualitative proof of the Simon-Ando theorem.

     

Class numbers and Iwasawa invariants of quadratic fields

Let and be quadratic fields with 2 (mod 3) a positive integer. Let be the respective Iwasawa -invariants of the cyclotomic -extension of these fields. We show that if , then 3 does not divide the class number of and .

     

On purely inseparable extensions and their generators

Let be a field of characteristic and a polynomial algebra in two variables. By a -generator of we mean an element of for which there exist and such that . We also define a -line of to mean any element of whose coordinate ring is that of a -generator. Then we prove that if is such that is a -line of (where is an indeterminate over ), then is a -generator of . This is analogous to the well-known fact that if is such that is a line of , then is a variable of . We also prove that if is a -line of for which there exist and such that , then is in fact a -generator of .