Multipliers of weighted spaces and reflexivity property

We prove for some translation-invariant weighted spaces the following characterization: is a multiplier of if and only if leaves invariant every translation-invariant subspace of . This result is equivalent with the reflexivity of the multiplier algebra of .

     

A note on meromorphic operators

Let be a complex Banach space and a bounded linear operator on . is called meromorphic if the spectrum of is a countable set, with the only possible point of accumulation, such that all the nonzero points of are poles of . By means of the analytical core we give a spectral theory of meromorphic operators. Our results are a generalization of some results obtained by Gong and Wang (2003).

     

A variant of the Reynolds operator

Let be a linearly reductive group over a field , and let be a -algebra with a rational action of . Given rational --modules and , we define for the induced -action on Hom a generalized Reynolds operator, which exists even if the action on Hom is not rational. Given an -module homomorphism , it produces, in a natural way, an -module homomorphism which is -equivariant. We use this generalized Reynolds operator to study properties of rational - modules. In particular, we prove that if is invariantly generated (i.e. ), then is a projective (resp. flat) -module provided that is a projective (resp. flat) -module. We also give a criterion whether an -projective (or -flat) rational --module is extended from an -module.

     

Index of B-Fredholm operators and generalization of a Weyl theorem

The aim of this paper is to show that if and are commuting B-Fredholm operators acting on a Banach space , then is a B-Fredholm operator and , where means the index. Moreover if is a B-Fredholm operator and is a finite rank operator, then is a B-Fredholm operator and We also show that if is isolated in the spectrum of , then is a B-Fredholm operator of index if and only if is Drazin invertible. In the case of a normal bounded linear operator acting on a Hilbert space , we obtain a generalization of a classical Weyl theorem.     

On restricted weak type : The continuous case

Let denote the unit circle. An example of a sublinear translation-invariant operator acting on is given such that is of restricted weak type but not of weak type .

     

Discrete spectra of -algebras and orthogonally closed submodules in Hilbert -modules

Let and be -algebras and let be an --imprimitivity bimodule. Then it is shown that if the spectrum of (resp. of ) is discrete, then every closed --submodule of is orthogonally closed in , and conversely that if (resp. ) is a -space and if every closed --submodule of is orthogonally closed in , then (resp. ) is discrete.

     

A pure subalgebra of a finitely generated algebra is finitely generated

We prove the following. Let be a Noetherian commutative ring, a finitely generated -algebra, and a pure -subalgebra of . Then is finitely generated over .

     

On isomorphisms of algebras of smooth functions

We show that for any smooth Hausdorff manifolds and , which are not necessarily second-countable, paracompact or connected, any isomorphism from the algebra of smooth (real or complex) functions on to the algebra of smooth functions on is given by composition with a unique diffeomorphism from to . An analogous result holds true for isomorphisms of algebras of smooth functions with compact support.

     

Uncountable categoricity for gross models

A model is said to be gross if all infinite definable sets in have the same cardinality (as ). We prove that if for some uncountable , has a unique gross model of cardinality , then for any uncountable , has a unique gross model of cardinality .

     

Normal bases for Hopf-Galois algebras

Let be a Hopf algebra over a commutative ring such that is a finitely generated, projective module over , let be a right -comodule algebra, and let be the subalgebra of -coinvariant elements of . If is a Galois extension of and is a local subalgebra of the center of , then is a cleft right -comodule algebra or, equivalently, there is a normal basis for over .

     

A characterization of the projective line

Let be a set (with at least three different points) and let be a group of bijections of . If the action of on satisfies three natural conditions, then admits a canonical structure of a projective line over a commutative field, such that is the group of all projective transformations of .

     

On the completeness of factor rings

Let be a complete local domain containing the integers with maximal ideal such that is at least the cardinality of the real numbers. Let be a nonmaximal prime ideal of such that is a regular local ring. We construct an excellent local ring such that the completion of is , the generic formal fiber of is local with maximal ideal and if is a nonzero ideal of , then is complete.

     

Normal essential eigenvalues in the boundary of the numerical range

A purely geometric property of a point in the boundary of the numerical range of an operator on Hilbert space is examined which implies that such a point is the value at of a multiplicative linear functional of the -algebra, , generated by and the identity operator. Roughly speaking, such a property means that the boundary of the numerical range (of ) has infinite curvature at that point. Furthermore, it is shown that if such a point is not a sharp linear corner of the numerical range of , then the multiplicative linear functional vanishes on the compact operators in .

     

Remarks concerning linear characters of reflection groups

Let be a finite group generated by unitary reflections in a Hermitian space , and let be a root of unity. Let be a subspace of , maximal with respect to the property of being a -eigenspace of an element of , and let be the parabolic subgroup of elements fixing pointwise. If is any linear character of , we give a condition for the restriction of to to be trivial in terms of the invariant theory of , and give a formula for the polynomial , where is the dimension of the -eigenspace of . Applications include criteria for regularity, and new connections between the invariant theory and the structure of .

     

Spectrum preserving linear mappings for scattered Jordan-Banach algebras

Given two semisimple complex Jordan-Banach algebras with identity and , we say that is a spectrum preserving linear mapping from to if is surjective and we have , for all . We prove that if is a scattered Jordan-Banach algebra, then is a Jordan isomorphism.

     

The linear space of generalized Brownian motions with applications

In this paper, we define, motivated by recent works of Chang and Skoug, stochastic integrals for a generalized Brownian motion ( ) and then use it to study the representation problem on the linear space spanned by . We next establish a translation theorem for -functionals of , , and then use this translation to establish an integration by parts formula for -functionals of .

     

Strong pseudo-contractions perturbed by compact operators in Banach spaces

Let be a (real) Banach space, let be an open subset of , and let denote the collection of all nonempty bounded and closed subsets of . Suppose is continuous from into with respect to the Hausdorff metric and strongly pseudo-contractive, while is compact from into . Then has a fixed point if it satisfies the classical Leray-Schauder condition on the boundary of .

     

Properly -realizable groups

A finitely presented group is said to be properly -realizable if there exists a compact -polyhedron with and whose universal cover has the proper homotopy type of a (p.l.) -manifold with boundary. In this paper we show that, after taking wedge with a -sphere, this property does not depend on the choice of the compact -polyhedron with . We also show that (i) all -ended and -ended groups are properly -realizable, and (ii) the class of properly -realizable groups is closed under amalgamated free products (HNN-extensions) over a finite cyclic group (as a step towards proving that -ended groups are properly -realizable, assuming -ended groups are).

     

Hopf algebras of dimension

Let be a finite-dimensional Hopf algebra over an algebraically closed field of characteristic 0. If is not semisimple and for some odd integer , then or is not unimodular. Using this result, we prove that if for some odd prime , then is semisimple. This completes the classification of Hopf algebras of dimension .

     

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.