Spectral structure and subdecomposability of -hyponormal operators

We prove that for every -hyponormal operator there corresponds a hyponormal operator such that and have ``equal spectral structure". We also prove that every -hyponormal operator is subdecomposable. Then some relevant quasisimilarity results are obtained, including that two quasisimilar -hyponormal operators have equal essential spectra.

     

Higher order symmetric spaces and the roots of the identity in a Lie group

Let denote the set of all -roots of the identity in a Lie group . We show that is always an embedded submanifold of , having the conjugacy classes of its elements as open submanifolds. These conjugacy classes are examples of -symmetric spaces and we show, more generally, that every -symmetric space of a Lie group is a covering manifold of an embedded submanifold of . We compute also the Hessian of the inclusions of and into , relative to the natural connection on the domain and to the symmetric connection on .

     

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 .

     

Schreier theorem on groups which split over free abelian groups

Let be either a free product with amalgamation or an HNN group where is isomorphic to a free abelian group of finite rank. Suppose that both and have no nontrivial, finitely generated, normal subgroups of infinite indices. We show that if contains a finitely generated normal subgroup which is neither contained in nor free, then the index of in is finite. Further, as an application of this result, we show that the fundamental group of a torus sum of -manifolds and , the interiors of which admit hyperbolic structures, have no nontrivial, finitely generated, nonfree, normal subgroup of infinite index if each of and has at least one nontorus boundary.

     

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 .

     

A description of Hilbert -modules in which all closed submodules are orthogonally closed

Let , be -algebras and a full Hilbert --bimodule such that every closed right submodule is orthogonally closed, i.e., . Then there are families of Hilbert spaces , such that and are isomorphic to -direct sums , resp. , and is isomorphic to the outer direct sum .

     

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 .

     

A generalization of Kelley's theorem for -spaces

We prove that if an open map of compacta and has perfect fibers and is a -space, then there exists a -dimensional compact subset of intersecting each fiber of . This is a stronger version of a well-known theorem of Kelley. Applications of this result and related topics are discussed.

     

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 .

     

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.

     

-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 .

     

Reduction in principal fiber bundles: Covariant Euler-Poincaré equations

Let be a principal -bundle, and let be a -invariant Lagrangian density. We obtain the Euler-Poincaré equations for the reduced Lagrangian defined on , the bundle of connections on .

     

Compactifications and universal spaces in extension theory

We show that for each countable simplicial complex the following conditions are equivalent:

• iff for any space .
• There exists a -invertible map of a metrizable compactum with onto the Hilbert cube.

     

Extensions of holomorphic maps through hypersurfaces and relations to the Hartogs extensions in infinite dimension

A generalization of Kwack's theorem to the infinite dimensional case is obtained. We consider a holomorphic map from into , where is a hypersurface in a complex Banach manifold and is a hyperbolic Banach space. Under various assumptions on , and we show that can be extended to a holomorphic map from into . Moreover, it is proved that an increasing union of pseudoconvex domains containing no complex lines has the Hartogs extension property.

     

Integer-valued polynomials over Krull-type domains and Prüfer -multiplication domains

Let be a domain with quotient field . The ring of integer-valued polynomials over is . We characterize the Krull-type domains such that is a Prüfer -multiplication domain.

     

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).

     

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 .

     

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.

     

On pointed Hopf algebras of dimension

In this note we describe nonsemisimple Hopf algebras of dimension with coradical isomorphic to , abelian of order , over an algebraically closed field of characteristic zero. If is cyclic or , then we also determine the number of isomorphism classes of such Hopf algebras.