On conjugation invariants in the dual Steenrod algebra

We investigate the canonical conjugation, , of the mod dual Steenrod algebra, , with a view to determining the subspace, , of elements invariant under . We give bounds on the dimension of this subspace for each degree and show that, after inverting , it becomes polynomial on a natural set of generators. Finally we note that, without inverting , is far from being polynomial.

     

Operators with bounded conjugation orbits

For a bounded invertible operator on a complex Banach space let be the set of operators in for which Suppose that and is in A bound is given on in terms of the spectral radius of the commutator. Replacing the condition in by the weaker condition as for every 0$">, an extension of the Deddens-Stampfli-Williams results on the commutant of is given.

     

A proof of W. T. Gowers' theorem

W. T. Gowers' theorem asserts that for every Lipschitz function and 0$">, there exists an infinite-dimensional subspace of such that the oscillation of on is at most . The proof of this theorem has been reduced by W. T. Gowers to the proof of a new Ramsey type theorem. Our aim is to present a proof of the last result.

     

Pure Picard-Vessiot extensions with generic properties

Given a connected linear algebraic group over an algebraically closed field of characteristic 0, we construct a pure Picard-Vessiot extension for , namely, a Picard-Vessiot extension , with differential Galois group , such that and are purely differentially transcendental over . The differential field is the quotient field of a -stable proper differential subring with the property that if is any differential field with field of constants and is a Picard-Vessiot extension with differential Galois group a connected subgroup of , then there is a differential homomorphism such that is generated over as a differential field by .

     

Algebraic isomorphisms and -subspace lattices

The class of -lattices was originally defined in the second author's thesis and subsequently by Longstaff, Nation, and Panaia. A subspace lattice on a Banach space which is also a -lattice is called a -subspace lattice, abbreviated JSL. It is demonstrated that every single element of has rank at most one. It is also shown that has the strong finite rank decomposability property. Let and be subspace lattices that are also JSL's on the Banach spaces and , respectively. The two properties just referred to, when combined, show that every algebraic isomorphism between and preserves rank. Finally we prove that every algebraic isomorphism between and is quasi-spatial.

     

Character degree sets that do not bound the class of a -group

Suppose that we are given a set of powers of a prime and that . A technique is presented that enables the construction of a -group of specified nilpotence class such that its set of irreducible character degrees is exactly . If , then this can be done for and if , then the only requirement is .

     

Are covering (enveloping) morphisms minimal?

We prove that for certain classes of modules such that direct sums of -covers ( -envelopes) are -covers ( -envelopes), -covering ( -enveloping) homomorphisms are always right (left) minimal. As a particular case we see that over noetherian rings, essential monomorphisms are left minimal. The same type of results are given when direct products of -covers are -covers. Finally we prove that over commutative noetherian rings, any direct product of flat covers of modules of finite length is a flat cover.     

On local irreducibility of the spectrum

Let be the algebra of complex matrices, and for denote by and the spectrum and spectral radius of respectively. Let be a domain in containing 0, and let be a holomorphic map. We prove: (1) if for , then for ; (2) if for , then again for . Both results are special cases of theorems expressing the irreducibility of the spectrum near .

     

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 dichotomy theorem for subsets of the power set of the natural numbers

The following dichotomy is established for any pair , of hereditary families of finite subsets of : Given , an infinite subset of , there exists an infinite subset of so that either , or , where denotes the set of all finite subsets of .

     

Hilbert modules over a class of semicrossed products

Given the disk algebra and an automorphism , there is associated a non-self-adjoint operator algebra called the semicrossed product of with . Buske and Peters showed that there is a one-to-one correspondence between the contractive Hilbert modules over and pairs of contractions and on satisfying . In this paper, we show that the orthogonally projective and Shilov Hilbert modules over correspond to pairs of isometries on satisfying . The problem of commutant lifting for is left open, but some related results are presented.     

Maximum curves and isolated points of entire functions

Given , curves belonging to the set of points were defined by Hardy to be maximum curves. Clunie asked the question as to whether the set could also contain isolated points. This paper shows that maximum curves consist of analytic arcs and determines a necessary condition for such curves to intersect. Given two entire functions and , if the maximum curve of is the real axis, conditions are found so that the real axis is also a maximum curve for the product function . By means of these results an entire function of infinite order is constructed for which the set has an infinite number of isolated points. A polynomial is also constructed with an isolated point.

     

Isomorphisms of subalgebras of nest algebras

Let be a subalgebra of a nest algebra . If contains all rank one operators in , then is said to be large; if the set of rank one operators in coincides with that in the Jacobson radical of , is said to be radical-type. In this paper, algebraic isomorphisms of large subalgebras and of radical-type subalgebras are characterized. Let be a nest of subspaces of a Hilbert space and be a subalgebra of the nest algebra associated to (). Let be an algebraic isomorphism from onto . It is proved that is spatial if one of the following occurs: (1) () is large and contains a masa; (2) () is large and closed; (3) () is a closed radical-type subalgebra and ( is quasi-continuous (i.e. the trivial elements of are limit points); (4) () is large and one of and is not quasi-continuous.

     

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 .

     

A lower bound for the number of components of the moduli schemes of stable rank 2 vector bundles on projective 3-folds

Fix a smooth projective 3-fold , , with ample, and . Assume the existence of integers with such that is numerically equivalent to . Let be the moduli scheme of -stable rank 2 vector bundles, , on with and . Let be the number ofits irreducible components. Then .

     

Perturbations and Weyl's theorem

A Banach space operator is completely hereditarily normaloid, , if either every part, and (also) for every invertible part , of is normaloid or if for every complex number every part of is normaloid. Sufficient conditions for the perturbation of by an algebraic operator to satisfy Weyl's theorem are proved. Our sufficient conditions lead us to the conclusion that the conjugate operator satisfies -Weyl's theorem.

     

Holomorphic perturbation of Fourier coefficients

Let be the unit circle, let be a Banach space continuously embedded in and suppose that is a Banach -module under convolution. We show that if and is holomorphic in a neighbourhood of with and then

     

Dense subsets of maximally almost periodic groups

A (discrete) group is said to be maximally almost periodic if the points of are distinguished by homomorphisms into compact Hausdorff groups. A Hausdorff topology on a group is totally bounded if whenever there is such that . For purposes of this abstract, a family with a totally bounded topological group is a strongly extraresolvable family if (a)  \vert G\vert$">, (b) each is dense in , and (c) distinct satisfy ; a totally bounded topological group with such a family is a strongly extraresolvable topological group.

We give two theorems, the second generalizing the first.

Theorem 1. Every infinite totally bounded group contains a dense strongly extraresolvable subgroup.

Corollary. In its largest totally bounded group topology, every infinite Abelian group is strongly extraresolvable.

Theorem 2. Let be maximally almost periodic. Then there are a subgroup of and a family such that

(i) is dense in every totally bounded group topology on ;

(ii) the family is a strongly extraresolvable family for every totally bounded group topology on such that ; and

(iii) admits a totally bounded group topology as in (ii).

Remark. In certain cases, for example when is Abelian, one must in Theorem 2 choose . In certain other cases, for example when the largest totally bounded group topology on is compact, the choice is impossible.

     

Double exponential sums over thin sets

We estimate double exponential sums of the form

where is of multiplicative order modulo the prime and and are arbitrary subsets of the residue ring modulo . In the special case , our bound is nontrivial for with any fixed 0$">, while if in addition we have it is nontrivial for .

     

Self-commutators of automorphic composition operators on the Dirichlet space

Let be a conformal automorphism on the unit disk and be the composition operator on the Dirichlet space induced by . In this article we completely determine the point spectrum, spectrum, essential spectrum and essential norm of the operators and self-commutators of , which expose that the spectrum and point spectrum coincide. We also find the eigenfunctions of the operators.