Global properties of the lattice of classes

Let be the lattice of classes of reals. We show there are exactly two possible isomorphism types of end intervals, . Moreover, finiteness is first order definable in .

     

Real rank and squaring mappings for unital -algebras

It is proved that if is a compact Hausdorff space of Lebesgue dimension , then the squaring mapping , defined by , is open if and only if . Hence the Lebesgue dimension of can be detected from openness of the squaring maps . In the case it is proved that the map , from the selfadjoint elements of a unital -algebra into its positive elements, is open if and only if is isomorphic to for some compact Hausdorff space with .

     

Lifts of - and -morphisms to -morphisms

Let be the Hochschild complex of cochains on and let be the space of multivector fields on . In this paper we prove that given any -structure (i.e. Gerstenhaber algebra up to homotopy structure) on , and any -morphism (i.e. morphism of a commutative, associative algebra up to homotopy) between and , there exists a -morphism between and that restricts to . We also show that any -morphism (i.e. morphism of a Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a -morphism, using Tamarkin's method for any -structure on . We also show that any two of such -morphisms are homotopic.

     

(APD)-property of -algebras by extensions of -algebras with (APD)

A unital -algebra is said to have the (APD)-property if every nonzero element in has the approximate polar decomposition. Let be a closed ideal of . Suppose that and have (APD). In this paper, we give a necessary and sufficient condition that makes have (APD). Furthermore, we show that if and or is a simple purely infinite -algebra, then has (APD).

     

On the error term in an asymptotic formula for the symmetric square -function

Recently Wu proved that for all primes ,

where runs over all normalized newforms of weight 2 and level . Here we show that can be replaced by .

     

Vector measure Banach spaces containing a complemented copy of

Let a Banach space and a -algebra of subsets of a set . We say that a vector measure Banach space has the bounded Vitaly-Hahn-Sacks Property if it satisfies the following condition: Every vector measure , for which there exists a bounded sequence in verifying for all , must belong to . Among other results, we prove that, if is a vector measure Banach space with the bounded V-H-S Property and containing a complemented copy of , then contains a copy of .

     

Cancellation of direct sums of countable abelian -groups

Let be abelian groups where is a direct sum of countable -groups. A condition is given on the Ulm-Kaplansky -invariants of and such that .

     

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.

     

An optimal Poincaré inequality in for convex domains

For convex domains with diameter we prove

for any with zero mean value on . We also show that the constant in this inequality is optimal.

     

Polynomial approximation on real-analytic varieties in

We prove: Let be a compact real-analytic variety in . Assume (i) is polynomially convex and (ii) every point of is a peak point for . Then . This generalizes a previous result of the authors on polynomial approximation on three-dimensional real-analytic submanifolds of .

     

A note on the support of a Sobolev function on a -cell

It is shown that a -cell (the homeomorphic image of a closed ball in ) in , , cannot support a function in if [\frac{k+1}{2}]$">, the greatest integer in .

     

-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 .

     

On the invariance of classes under composition

The necessary and sufficient condition for to be in the class for every of that class whose range is in the domain of is that be in .

     

On an approximate automorphism on a -algebra

It is shown that for an approximate algebra homomorphism on a Banach -algebra , there exists a unique algebra -homomorphism near the approximate algebra homomorphism. This is applied to show that for an approximate automorphism on a unital -algebra , there exists a unique automorphism near the approximate automorphism. In fact, we show that the approximate automorphism is an automorphism.

     

On the best possible character of the norm in some a priori estimates for non-divergence form equations in Carnot groups

Let be a group of Heisenberg type with homogeneous dimension . For every we construct a non-divergence form operator and a non-trivial solution to the Dirichlet problem: in , on . This non-uniqueness result shows the impossibility of controlling the maximum of with an norm of when . Another consequence is the impossiblity of an Alexandrov-Bakelman type estimate such as

where is the dimension of the horizontal layer of the Lie algebra and is the symmetrized horizontal Hessian of .

     

Solution to a problem of S. Payne

A problem posed by S. Payne calls for determination of all linearized polynomials such that and are permutations of and respectively. We show that such polynomials are exactly of the form with and . In fact, we solve a -ary version of Payne's problem.

     

A concrete description of -spaces as -spaces and its applications

We prove that for a compact Hausdorff space without isolated points, and are isometrically Riesz isomorphic spaces under a certain topology on . Moreover, is a closed subspace of . This provides concrete examples of compact Hausdorff spaces such that the Dedekind completion of is (= the set of all bounded real-valued functions on ) since the Dedekind completion of is ( and spaces as Banach lattices).

     

On Berry-Esseen bounds of summability transforms

Let , , , be a collection of random variables, where for each , , , are independent. Let be a regular summability method. We provide some rates of convergence (Berry-Esseen type bounds) for the weak convergence of summability transform . We show that when is the classical Cesáro summability method, the rate of convergence of the resulting central limit theorem is best possible among all regular triangular summability methods with rows adding up to one. We further provide some summability results concerning -negligibility. An application of these results characterizes the rate of convergence of Schnabl operators while approximating Lipschitz continuous functions.

     

The Russo-Dye Theorem in a weakly closed -module

Suppose that is admissible. It is shown that the convex hull of unitary elements of a weakly closed -module contains the whole unit ball of if and only if and for any 0$">, 0$">.

     

Automatic continuity of -derivations on -algebras

Let be a -algebra acting on a Hilbert space , let be a linear mapping and let be a -derivation. Generalizing the celebrated theorem of Sakai, we prove that if is a continuous -mapping, then is automatically continuous. In addition, we show the converse is true in the sense that if is a continuous --derivation, then there exists a continuous linear mapping such that is a --derivation. The continuity of the so-called - -derivations is also discussed.