-summand vectors in Banach spaces

The aim of this paper is to study the set of all -summand vectors of a real Banach space . We provide a characterization of -summand vectors in smooth real Banach spaces and a general decomposition theorem which shows that every real Banach space can be decomposed as an -sum of a Hilbert space and a Banach space without nontrivial -summand vectors. As a consequence, we generalize some results and we obtain intrinsic characterizations of real Hilbert spaces.

     

The sufficiency of arithmetic progressions for the Conjecture

Define by if is odd and if is even. The Conjecture states that the -orbit of every positive integer contains . A set of positive integers is said to be sufficient if the -orbit of every positive integer intersects the -orbit of an element of that set. Thus to prove the Conjecture it suffices to prove it on some sufficient set. Andaloro proved that the sets are sufficient for and asked if is also sufficient for larger values of . We answer this question in the affirmative by proving the stronger result that is sufficient for any nonnegative integers and with i.e. every nonconstant arithmetic sequence forms a sufficient set. We then prove analagous results for the Divergent Orbits Conjecture and Nontrivial Cycles Conjecture.

     

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.

     

An ultrafilter with property

Let be a -supercompact cardinal. We show that carries a normal ultrafilter with a property introduced by Menas. With it we give a transparent proof of Kamo's theorem that carries a normal ultrafilter with the partition property.

     

Transitive families of projections in factors of type

We introduce a notion of transitive family of subspaces relative to a type factor, and hence a notion of transitive family of projections in such a factor. We show that whenever is a factor of type and is generated by two self-adjoint elements, then contains a transitive family of projections. Finally, we exhibit a free transitive family of projections that generate a factor of type .

     

All -cotilting modules are pure-injective

We prove that all -cotilting -modules are pure-injective for any ring and any . To achieve this, we prove that is a covering class whenever is an -module such that is closed under products and pure submodules.

     

On maximal operators on -spheres in

A. Magyar's result on -bounds for a family of operators on -spheres () in is improved to match the corresponding theorem for -spheres.

     

Global dominated splittings and the Newhouse phenomenon

We prove that given a compact -dimensional boundaryless manifold , , there exists a residual subset of the space of diffeomorphisms such that given any chain-transitive set of , then either admits a dominated splitting or else is contained in the closure of an infinite number of periodic sinks/sources. This result generalizes the generic dichotomy for homoclinic classes given by Bonatti, Diaz, and Pujals (2003).

It follows from the above result that given a -generic diffeomorphism , then either the nonwandering set may be decomposed into a finite number of pairwise disjoint compact sets each of which admits a dominated splitting, or else exhibits infinitely many periodic sinks/sources (the `` Newhouse phenomenon"). This result answers a question of Bonatti, Diaz, and Pujals and generalizes the generic dichotomy for surface diffeomorphisms given by Mañé (1982).

     

On chaotic -semigroups and infinitely regular hypercyclic vectors

A -semigroup on a Banach space is called hypercyclic if there exists an element such that is dense in . is called chaotic if is hypercyclic and the set of its periodic vectors is dense in as well. We show that a spectral condition introduced by Desch, Schappacher and Webb requiring many eigenvectors of the generator which depend analytically on the eigenvalues not only implies the chaoticity of the semigroup but the chaoticity of every . Furthermore, we show that semigroups whose generators have compact resolvent are never chaotic. In a second part we prove the existence of hypercyclic vectors in for a hypercyclic semigroup , where is its generator.

     

The universal central extension of the three-point loop algebra

We consider the three-point loop algebra,

where denotes a field of characteristic 0 and is an indeterminate. The universal central extension of was determined by Bremner. In this note, we give a presentation for via generators and relations, which highlights a certain symmetry over the alternating group . To obtain our presentation of , we use the realization of as the tetrahedron Lie algebra.

     

A problem of Baernstein on the equality of the -harmonic measure of a set and its closure

A. Baernstein II (Comparison of -harmonic measures of subsets of the unit circle, St. Petersburg Math. J. 9 (1998), 543-551, p. 548), posed the following question: If is a union of open arcs on the boundary of the unit disc , then is , where denotes the -harmonic measure? (Strictly speaking he stated this question for the case .) For the positive answer to this question is well known. Recall that for the -harmonic measure, being a nonlinear analogue of the harmonic measure, is not a measure in the usual sense.

The purpose of this note is to answer a more general version of Baernstein's question in the affirmative when . In the proof, using a deep trace result of Jonsson and Wallin, it is first shown that the characteristic function is the restriction to of a Sobolev function from .

For it is no longer true that belongs to the trace class. Nevertheless, we are able to show equality for the case of one arc for all , using a very elementary argument. A similar argument is used to obtain a result for starshaped domains.

Finally we show that in a certain sense the equality holds for almost all relatively open sets.

     

Local automorphisms and derivations on certain -algebras

It is shown that continuous -local derivations on -algebras are derivations and surjective -local *-automorphisms on prime -algebras or on -algebras such that the identity element is properly infinite are *-automorphisms.

     

Tensor products of -weakly closed nest algebra submodules

In this paper we prove that for any unital -weakly closed algebra which is -weakly generated by finite-rank operators in , every -weakly closed -submodule has . In the case of nest algebras, if are nests, we obtain the following -fold tensor product formula:

where each is the -weakly closed Alg -submodule determined by an order homomorphism from into itself.

     

A remark on irregularity of the -Neumann problem on non-smooth domains

It is an observation due to J. J. Kohn that for a smooth bounded pseudoconvex domain in there exists such that the -Neumann operator on maps (the space of -forms with coefficient functions in -Sobolev space of order ) into itself continuously. We show that this conclusion does not hold without the smoothness assumption by constructing a bounded pseudoconvex domain in , smooth except at one point, whose -Neumann operator is not bounded on for any .

     

There is no separable universal -factor

Gromov constructed uncountably many pairwise nonisomorphic discrete groups with Kazhdan's property . We will show that no separable -factor can contain all these groups in its unitary group. In particular, no separable -factor can contain all separable -factors in it. We also show that the full group -algebras of some of these groups fail the lifting property.

     

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 .

     

On characterization and perturbation of local -semigroups

Let be a -group with generator , and let be a local -semigroup commuting with . Then the operators , , form a local -semigroup. It is proved that if is injective and is the generator of , then is closable and is the generator of . Also proved are a characterization theorem for local -semigroups with not necessarily injective and a theorem about solvability of the abstract inhomogeneous Cauchy problem:

     

Functions -orthogonal with respect to their own zeros

In 1939, G. H. Hardy proved that, under certain conditions, the only functions satisfying

where the are the zeros of , are the Bessel functions. We replace the above integral by the Jackson -integral and give the -analogue of Hardy's result.

     

Zero distribution of Müntz extremal polynomials in

Let be a sequence of distinct positive numbers. Let and let denote the extremal Müntz polynomial in with exponents . We investigate the zero distribution of . In particular, we show that if

then the normalized zero counting measure of converges weakly as to

while if or , the limiting measure is a Dirac delta at or , respectively.

     

Basis properties of eigenfunctions of the -Laplacian

For , the eigenfunctions of the non-linear eigenvalue problem for the -Laplacian on the interval are shown to form a Riesz basis of and a Schauder basis of whenever .