A lifting theorem for symmetric commutants

Let be bounded operators on a Hilbert space such that . Given a symmetry on , i.e., , we define the -symmetric commutant of to be the operator space

In this paper we obtain lifting theorems for symmetric commutants. The result extends the Sz.-Nagy-Foias commutant lifting theorem (), the anticommutant lifting theorem of Sebestyén ( ), and the noncommutative commutant lifting theorem ( ). Sarason's interpolation theorem for is extended to symmetric commutants on Fock spaces.     

Minimal sufficiency of order statistics in convex models

Let be a convex and dominated statistical model on the measurable space , with minimal sufficient, and let . Then , the -algebra of all permutation invariant sets belonging to the -fold product -algebra , is shown to be minimal sufficient for the corresponding model for independent observations, .

The main technical tool provided and used is a functional analogue of a theorem of Grzegorek (1982) concerning generators of .

     

The Dirichlet-Jordan test and multidimensional extensions

If is a foliation of an open set by smooth -dimensional surfaces, we define a class of functions , supported in , that are, roughly speaking, smooth along and of bounded variation transverse to . We investigate geometrical conditions on that imply results on pointwise Fourier inversion for these functions. We also note similar results for functions on spheres, on compact 2-dimensional manifolds, and on the 3-dimensional torus. These results are multidimensional analogues of the classical Dirichlet-Jordan test of pointwise convergence of Fourier series in one variable.

     

Spaces of type and the Hankel convolution

In this paper we introduce new function spaces that are denoted by , -1/2$"> and and that are spaces of type where the Hankel convolution and the Hankel transformation are defined. The spaces will play the same role in the Hankel setting that the spaces play in the theory of Fourier transformation.     

An order characterization of commutativity for -algebras

In this paper, we investigate the problem of when a -algebra is commutative through operator-monotonic increasing functions. The principal result is that the function is operator-monotonic increasing on a -algebra if and only if is commutative. Therefore, -algebra is commutative if and only if in for all positive elements in .

     

On the Betti numbers of sign conditions

Let be a real closed field and let and be finite subsets of such that the set has elements, the algebraic set defined by has dimension and the elements of and have degree at most . For each we denote the sum of the -th Betti numbers over the realizations of all sign conditions of on by . We prove that

This generalizes to all the higher Betti numbers the bound on . We also prove, using similar methods, that the sum of the Betti numbers of the intersection of with a closed semi-algebraic set, defined by a quantifier-free Boolean formula without negations with atoms of the form or for , is bounded by

making the bound more precise.

     

Types in reductive -adic groups: The Hecke algebra of a cover

In this paper, is a non-Archimedean local field and is the group of -points of a connected reductive algebraic group defined over . Also, is an irreducible representation of a compact open subgroup of , the pair being a type in . The pair is assumed to be a cover of a type in a Levi subgroup of . We give conditions, generalizing those of earlier work, under which the Hecke algebra is the tensor product of a canonical image of and a sub-algebra , for a compact open subgroup of containing .

     

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

     

The distribution of solutions of the congruence

For a cube of size , we obtain a lower bound on so that is nonempty, where is the algebraic subset of defined by

a positive integer and an integer not divisible by . For we obtain that is nonempty if , for we obtain that is nonempty if , and for we obtain that is nonempty if . Using the assumption of the Grand Riemann Hypothesis we obtain is nonempty if .

     

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.

     

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 .

     

Compact range property and operators on -algebras

We prove that a Banach space has the compact range property (CRP) if and only if, for any given -algebra , every absolutely summing operator from into is compact. Related results for -summing operators () are also discussed as well as operators on non-commutative -spaces and -summing operators.

     

On the commutant of operators of multiplication by univalent functions

Let be a certain Banach space consisting of continuous functions defined on the open unit disk. Let be a univalent function defined on , and assume that denotes the operator of multiplication by . We characterize the structure of the operator such that . We show that for some function in . We also characterize the commutant of under certain conditions.

     

Convex curves, Radon transforms and convolution operators defined by singular measures

Let be a convex curve in the plane and let be the arc-length measure of Let us rotate by an angle and let be the corresponding measure. Let . Then This is optimal for an arbitrary . Depending on the curvature of , this estimate can be improved by introducing mixed-norm estimates of the form where and are conjugate exponents.     

On -good module categories without short cycles

Let be a quasi-hereditary algebra, and the -good module category consisting of -modules which have a filtration by standard modules. An indecomposable module in is said to be on a short cycle in if there exist an indecomposable module in and a chain of two nonzero noninvertible maps . It is shown that two indecomposable modules in are isomorphic if they are not on short cycles in and have the same composition factors. Moreover, if there is no short cycle in , we show that is finite, that is, there are only finitely many isomorphism classes of indecomposables in . This is an analogue to a result in a complete module category proved by Happel and Liu.

     

Blocks of central -group extensions

Let and be finite groups that have a common central -subgroup for a prime number , and let and respectively be -blocks of and induced by -blocks and respectively of and , both of which have the same defect group. We prove that if and are Morita equivalent via a certain special -bimodule, then such a Morita equivalence lifts to a Morita equivalence between and .

     

Helly-type theorems for homothets of planar convex curves

Helly's theorem implies that if is a finite collection of (positive) homothets of a planar convex body , any three having non-empty intersection, then has non-empty intersection. We show that for collections of homothets (including translates) of the boundary , if any four curves in have non-empty intersection, then has non-empty intersection. We prove the following dual version: If any four points of a finite set in the plane can be covered by a translate [homothet] of , then can be covered by a translate [homothet] of . These results are best possible in general.

     

Average values of symmetric square -functions at the edge of the critical strip

Let be the set of all normalized newforms of weight 2 and level , and let be the symmetric square -function associated to . If is a prime, then there is a positive constant such that

This improves a recent result of Akbary, which requires in place 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 .

     

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.