Functional calculus and -regularity of a class of Banach algebras

Suppose that is a -dynamical system such that is of polynomial growth. If is finite dimensional, we show that any element in has slow growth and that is -regular. Furthermore, if is discrete and is a ``nice representation' of , we define a new Banach -algebra which coincides with when is finite dimensional. We also show that any element in has slow growth and is -regular.

     

Witt kernels of bilinear forms for algebraic extensions in characteristic

Let be a field of characteristic and let be a purely inseparable extension of exponent . We determine the kernel of the natural restriction map between the Witt rings of bilinear forms of and , respectively. This complements a result by Laghribi who computed the kernel for the Witt groups of quadratic forms for such an extension . Based on this result, we will determine for a wide class of finite extensions which are not necessarily purely inseparable.

     

Failure of Krull-Schmidt for invertible lattices over a discrete valuation ring

Let be a prime greater than , and let be the semi-direct product of a group of order by a cyclic group of order , which acts faithfully on . Let be the localization of at . We show that the Krull-Schmidt Theorem fails for the category of invertible -lattices.

     

On stable equivalences induced by exact functors

Let and be two Artin algebras with no semisimple summands. Suppose that there is a stable equivalence between and such that is induced by exact functors. We present a nice correspondence between indecomposable modules over and . As a consequence, we have the following: (1) If is a self-injective algebra, then so is ; (2) If and are finite dimensional algebras over an algebraically closed field , and if is of finite representation type such that the Auslander-Reiten quiver of has no oriented cycles, then and are Morita equivalent.

     

Maps into complex space

If the dimension of is denoted by or , then a generic map satisfies , while in certain cases there is no map that satisfies .

     

Equicompact sets of operators defined on Banach spaces

Let and be Banach spaces. We say that a set denotes the space of all compact operators from into ) is equicompact if there exists a null sequence in such that for all and all . It is easy to show that collectively compactness and equicompactness are dual concepts in the following sense: is equicompact iff is collectively compact. We study some properties of equicompact sets and, among other results, we prove: 1) a set is equicompact iff each bounded sequence in has a subsequence such that is a converging sequence uniformly for ; 2) if does not have finite cotype and is a maximal equicompact set, then, given and a finite set in , there is an operator such that for and all .

     

A characterisation of in terms of forcing

We show that ``saturation' of the universe with respect to forcing over with partial orders on is equivalent to the existence of .

     

for commutative rings with identity

Let , , , , be the usual operators on classes of rings: and for isomorphic and homomorphic images of rings and , , respectively for subrings, direct, and subdirect products of rings. If is a class of commutative rings with identity (and in general of any kind of algebraic structures), then the class is known to be the variety generated by the class . Although the class is in general a proper subclass of the class for many familiar varieties . Our goal is to give an example of a class of commutative rings with identity such that . As a consequence we will describe the structure of two partially ordered monoids of operators.

     

Fiber products, Poincaré duality and -ring spectra

For a Poincaré duality space and a map , consider the homotopy fiber product . If is orientable with respect to a multiplicative cohomology theory , then, after suitably regrading, it is shown that the -homology of has the structure of a graded associative algebra. When is the diagonal map of a manifold , one recovers a result of Chas and Sullivan about the homology of the unbased loop space .

     

Projections in operator ranges

If is a Hilbert space, is a positive bounded linear operator on and is a closed subspace of , the relative position between and establishes a notion of compatibility. We show that the compatibility of is equivalent to the existence of a convenient orthogonal projection in the operator range with its canonical Hilbertian structure.

     

Eigenvalues of the Laplacian acting on -forms and metric conformal deformations

Let be a compact connected orientable Riemannian manifold of dimension and let be the -th positive eigenvalue of the Laplacian acting on differential forms of degree on . We prove that the metric can be conformally deformed to a metric , having the same volume as , with arbitrarily large for all .

Note that for the other values of , that is and , one can deduce from the literature that, 0$">, the -th eigenvalue is uniformly bounded on any conformal class of metrics of fixed volume on .

For , we show that, for any positive integer , there exists a metric conformal to such that, , , that is, the first eigenforms of are all exact forms.

     

-bounded groups and other topological groups with strong combinatorial properties

We construct several topological groups with very strong combinatorial properties. In particular, we give simple examples of subgroups of (thus strictly -bounded) which have the Menger and Hurewicz properties but are not -compact, and show that the product of two -bounded subgroups of may fail to be -bounded, even when they satisfy the stronger property . This solves a problem of Tkacenko and Hernandez, and extends independent solutions of Krawczyk and Michalewski and of Banakh, Nickolas, and Sanchis. We also construct separable metrizable groups of size continuum such that every countable Borel -cover of contains a -cover of .

     

Lightface -indescribable cardinals

-absoluteness for forcing means that for any forcing , . `` inaccessible to reals' means that for any real , . To measure the exact consistency strength of `` -absoluteness for forcing and is inaccessible to reals', we introduce a weak version of a weakly compact cardinal, namely, a (lightface) -indescribable cardinal; has this property exactly if it is inaccessible and .

     

Separate continuity, joint continuity and the Lindelöf property

In this paper we prove a theorem more general than the following. Suppose that is Lindelöf and -favourable and is Lindelöf and Cech-complete. Then for each separately continuous function there exists a residual set in such that is jointly continuous at each point of .

     

Multiplication and division by inner functions in the space of Bloch functions

A subspace of the Hardy space is said to have the -property if whenever and is an inner function with . We let denote the space of Bloch functions and the little Bloch space. Anderson proved in 1979 that the space does not have the -property. However, the question of whether or not ( ) has the -property was open. We prove that for every the space does not have the -property.

We also prove that if is any infinite Blaschke product with positive zeros and is a Bloch function with , as , then the product is not a Bloch function.

     

Linear functionals on the Cuntz algebra

For a pure state on , which is an extension of a pure state on with the property that if is a corresponding representation, then , induces a unital shift of of the Powers index . We describe states on by using sequences of unit vectors in . We study the linear functionals on the Cuntz algebra whose restrictions are the product pure state on . We find conditions on the sequence of unit vectors for which the corresponding linear functionals on become states under these conditions.

     

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 .

     

Localization of injective modules over valuation rings

It is proved that is injective if is an injective module over a valuation ring , for each prime ideal . Moreover, if or is flat, then is injective, too. It follows that localizations of injective modules over h-local Prüfer domains are injective, too.

     

A class of differentiable toral maps which are topologically mixing

We show that on the 2-torus there exists a open set of regular maps such that every map belonging to is topologically mixing but is not Anosov. It was shown by Mañé that this property fails for the class of toral diffeomorphisms, but that the property does hold for the class of diffeomorphisms on the 3-torus . Recently Bonatti and Diaz proved that the second result of Mañé is also true for the class of diffeomorphisms on the -torus ().

     

Equivariant crystalline cohomology and base change

Given a perfect field of characteristic , a smooth proper -scheme , a crystal on relative to and a finite group acting on and , we show that, viewed as a virtual -module, the reduction modulo of the crystalline cohomology of is the de Rham cohomology of modulo . On the way we prove a base change theorem for the virtual -representations associated with -equivariant objects in the derived category of -modules.