Maximal estimates for the -dimensional Walsh-Fourier series

The -dimensional dyadic martingale Hardy spaces are introduced and it is proved that the maximal operator of the means of a Walsh-Fourier series is bounded from to and is of weak type , provided that the supremum in the maximal operator is taken over a positive cone. As a consequence we obtain that the means of a function converge a.e. to the function in question. Moreover, we prove that the means are uniformly bounded on whenever . Thus, in case , the means converge to in norm. The same results are proved for the conjugate means, too.

     

Annihilating a subspace of with the sign of a continuous function

Let be a -finite, nonatomic, Baire measure space. Let be a finite dimensional subspace of . There is a bounded, continuous function, , defined on , such that

(1) for all , and (2) almost everywhere.

     

Finite generation properties for fuchsian group von Neumann algebras tensor

We prove that the algebra , a free group with finitely many generators, contains a subnormal operator such that the linear span of the set is weakly dense in . This is the analogue for the factor , finite, of a well known fact about the unilateral shift on a Hilbert space : the linear span of all the monomials is weakly dense in .

We also show that for a suitable space of square summable analytic functions, if is the projection from the Hilbert space of all square summable functions onto and is the unbounded operator of multiplication by on , then the (unbounded) operator is nonzero (with nonzero domain).

     

Iwasawa invariants and class numbers of quadratic fields for the prime

Let be a square-free integer with and . Put and . For the cyclotomic -extension of , we denote by the -th layer of over . We prove that the -Sylow subgroup of the ideal class group of is trivial for all integers if and only if the class number of is not divisible by the prime . This enables us to show that there exist infinitely many real quadratic fields in which splits and whose Iwasawa -invariant vanishes.

     

Normalizers of the congruence subgroups of the Hecke group II

Let . Let be an ideal of and let be the maximal ideal of such that . Then . In particular, if is square free, then is self-normalized in .

     

Universal -lattices of minimal rank

Let be the minimal rank of -universal -lattices, by which we mean positive definite -lattices which represent all positive -lattices of rank . It is a well known fact that for . In this paper, we determine and find all -universal lattices of rank for .

     

-sequences

A sequence of positive integers is called a -sequence if every integer has at most representations with all in and . A -sequence is also called a -sequence or Sidon sequence. The main result is the following

Theorem. Let be a -sequence and for an integer . Then there is a -sequence of size , where .

Corollary. Let . The interval then contains a -sequence of size , when .

     

A criterion for splitting -algebras in tensor products

The goal of the paper is to prove the following theorem: if , are unital -algebras, simple and nuclear, then any -subalgebra of the -tensor product of and , which contains the tensor product of with the scalar multiples of the unit of , splits in the -tensor product of with some -subalgebra of .

     

The globally irreducible representations of symmetric groups

Let be an algebraic number field and be the ring of integers of . Let be a finite group and be a finitely generated torsion free -module. We say that is a globally irreducible -module if, for every maximal ideal of , the -module is irreducible, where stands for the residue field .

Answering a question of Pham Huu Tiep, we prove that the symmetric group does not have non-trivial globally irreducible modules. More precisely we establish that if is a globally irreducible -module, then is an -module of rank with the trivial or sign action of .

     

A note on -bases of rings

Let be a tower of rings of characteristic . Suppose that is a finitely presented -module. We give necessary and sufficient conditions for the existence of -bases of over . Next, let be a polynomial ring where is a perfect field of characteristic , and let be a regular noetherian subring of containing such that . Suppose that is a free -module. Then, applying the above result to a tower of rings, we shall show that a polynomial of minimal degree in is a -basis of over .

     

On a theorem of Barbara Schmid

Let be a finite group and a complex representation. Barbara Schmid has shown that the algebra of invariant polynomial functions on the vector space is generated by homogeneous polynomials of degree at most , where is the largest degree of a generator in a minimal generating set for , and is the complex regular representation of . In this note we give a new proof of this result, and at the same time extend it to fields whose characteristic is larger than , the order of the group .

     

Conjugate -connections and holonomy groups

In this article we show that when the structure group of the reducible principal bundle is and is an -subbundle of , the rank of the holonomy group of a connection which is gauge equivalent to its conjugate connection is less than or equal to , and use the estimate to show that for all odd prime , if the holonomy group of the irreducible connection as above is simple and is not isomorphic to , , or , then it is isomorphic to .

     

Relations between the Taylor spectrum and the Xia spectrum

Let be a doubly commuting -tuple of -hyponormal operators with unitary operators from the polar decompositions . Let and . In this paper, we will show relations between the Taylor spectrum and the Xia spectrum .

     

On semisimple Hopf algebras of dimension

We consider the problem of the classification of semisimple Hopf algebras of dimension where are two prime numbers. First we prove that the order of the group of grouplike elements of is not , and that if it is , then . We use it to prove that if and its dual Hopf algebra are of Frobenius type, then is either a group algebra or a dual of a group algebra. Finally, we give a complete classification in dimension , and a partial classification in dimensions and .

     

Every -regular

We prove the following: Theorem A. If is a -regular ultrafilter, then either

(a)

is -regular, or

(b)

the cofinality of the linear order is , and is -regular for all .

Corollary B. Suppose that is singular, and either is regular, or . Then every -regular ultrafilter is -regular.

We also discuss some consequences and variations.

     

A characterization of

This work is devoted to the relationship between topological properties of a space and those of (= the space of continuous real-valued functions on , with the topology of pointwise convergence). The emphasis is on -compactness of and on location of in . In particular, -compact cosmic spaces are characterized in this way.

     

Completely continuous multilinear operators on spaces

Given a -linear operator from a product of spaces into a Banach space , our main result proves the equivalence between being completely continuous, having an -valued separately continuous extension to the product of the biduals and having a regular associated polymeasure. It is well known that, in the linear case, these are also equivalent to being weakly compact, and that, for , being weakly compact implies the conditions above but the converse fails.

     

On covering translations and homeotopy groups of contractible open -manifolds

This paper gives a new proof of a result of Geoghegan and Mihalik which states that whenever a contractible open -manifold which is not homeomorphic to is a covering space of an -manifold and either or and is irreducible, then the group of covering translations injects into the homeotopy group of .

     

Tilting up iterated tilted algebras

We show that, if is a representation-finite iterated tilted algebra of euclidean type , then there exist a sequence of algebras , and a sequence of modules , where , such that each is an APR-tilting -module, or an APR-cotilting -module, and is tilted representation-finite.

     

No submaximal topology on a countable set is -complementary

Two -topologies and given on the same set , are called transversal if their union generates the discrete topology on . The topologies and are -complementary if they are transversal and their intersection is the cofinite topology on . We establish that for any connected Tychonoff topology there exists a connected Tychonoff transversal one. Another result is that no -complementary topology exists for the maximal topology constructed by van Douwen on the rational numbers. This gives a negative answer to Problem 162 from Open Problems in Topology (1990).