Representation of continuous functions as sums of Green functions

Let , where is polar and compact and is a domain with Green function . We characterize those subsets of which have the following property: Every positive continuous function on can be written as , where and for each .

     

Completely distributive CSL algebras with no complements in

Anoussis and Katsoulis have obtained a criterion for the space to have a closed complement in , where is a completely distributive commutative subspace lattice. They show that, for a given , the set of for which this complement exists forms an interval whose endpoints are harmonic conjugates. Also, they establish the existence of a lattice for which has no complement for any . However, they give no specific example. In this note an elementary demonstration of a simple example of this phenomenon is given. From this it follows that for a wide range of lattices , fails to have a complement for any .

     

Characterization of the Fourier series of a distribution having a value at a point

Let be a periodic distribution of period . Let be its Fourier series. We show that the distributional point value exists and equals if and only if the partial sums converge to in the Cesàro sense as for each .

     

The existence of a maximizing vector for the numerical range of a compact operator

Let be a complex Lebesgue space with a unique duality map from to , the conjugate space of . Let be a compact operator on . This paper focuses on properties of and . We adapt an example due to Halmos to show that for , there is a compact operator on with the semi-open interval . So is not attained as a maximum. A corollary of the main result in this paper is that if , and , then is attained as a maximum.

     

Finite and -resolvability

A topological space is -resolvable if has disjoint dense subsets. In this paper, we prove that if is -resolvable for each positive integer , then is -resolvable.

     

A completely regular space which is the -complement of itself

Two topologies and on a fixed set are -complements if is the cofinite topology and is a sub-base for the discrete topology. In 1967, Steiner and Steiner showed that of any two -complements on a countable set, at least one is not Hausdorff. In 1969, Anderson and Stewart asked whether a Hausdorff topology on an uncountable set can have a Hausdorff -complement. We construct two homeomorphic completely regular -complementary topologies.

     

Cohomological detection and regular elements in group cohomology

Suppose that is a compact Lie group or a discrete group of finite virtual cohomological dimension and that is a field of characteristic . Suppose that is a set of elementary abelian -subgroups such that the cohomology is detected on the centralizers of the elements of . Assume also that is closed under conjugation and that is in whenever some subgroup of is in . Then there exists a regular element in the cohomology ring such that the restriction of to an elementary abelian -subgroup is not nilpotent if and only if is in . The converse of the result is a theorem of Lannes, Schwartz and the second author. The results have several implications for the depth and associated primes of the cohomology rings.

     

Isometries of certain operator algebras

To a given basis on an -dimensional Hilbert space , we associate the algebra of all linear operators on having every as an eigenvector. So, is commutative, semisimple, and -dimensional. Given two algebras of this type, and , there is a natural algebraic isomorphism of and . We study the question: When does preserve the operator norm?

     

On Voiculescu's double commutant theorem

For a separable infinite-dimensional Hilbert space , we consider the full algebra of bounded linear transformations and the unique non-trivial norm-closed two-sided ideal of compact operators . We also consider the quotient -algebra with quotient map

For any -subalgebra of , the relative commutant is given by for all in . It was shown by D. Voiculescu that, for any separable unital -subalgebra of ,

In this note, we exhibit a non-separable unital -subalgebra of for which (VDCT) fails.

     

Interpolation spaces between the Lipschitz class and the space of continuous functions

It is shown that the complex interpolation spaces and do not coincide with or and also that the couple is not a Calderón couple. Similar results are also obtained for the couples and when .

     

On contravariant finiteness of subcategories of modules of projective dimension

Let be an artin algebra. This paper presents a sufficient condition for the subcategory of to be contravariantly finite in , where is the subcategory of consisting of --modules of projective dimension less than or equal to . As an application of this condition it is shown that is contravariantly finite in for each when is stably equivalent to a hereditary algebra.

     

Differences of vector-valued functions on topological groups

Let be a locally compact group equipped with right Haar measure. The right differences of functions on are defined by for . Let and suppose for some and all . We prove that is a right uniformly continuous function of . If is abelian and the Beurling spectrum does not contain the unit of the dual group , then we show . These results have analogues for functions , where is a separable or reflexive Banach space. Finally, we apply our methods to vector-valued right uniformly continuous differences and to absolutely continuous elements of left Banach -modules.

     

On a measure-theoretic problem of Arveson

A probability measure on a product space is said to be bistochastic with respect to measures on and on if the marginals and are exactly and . A solution is presented to a problem of Arveson about sets which are of measure zero for all such .

     

Bounds for the operator norms of some Nörlund matrices

Suppose is a non-increasing sequence of non-negative numbers with , , , and is the lower triangular matrix defined by , , and , . We show that the operator norm of as a linear operator on is no greater than , for ; this generalizes, yet again, Hardy's inequality for sequences, and simplifies and improves, in this special case, more generally applicable results of D. Borwein, Cass, and Kratz. When the tend to a positive limit, the operator norm of on is exactly . We also give some cases when the operator norm of on is less than .

     

A rigidity theorem for the Clifford tori in

Let be the unit hypersphere in the 4-dimensional Euclidean space defined by . For each with , we denote by the Clifford torus in given by the equations and . The Clifford torus is a flat Riemannian manifold equipped with the metric induced by the inclusion map . In this note we prove the following rigidity theorem: If is an isometric embedding, then there exists an isometry of such that . We also show no flat torus with the intrinsic diameter is embeddable in except for a Clifford torus.

     

A topological characterization of linearity for quasi-traces

Let be a -algebra, and let be a (local) quasi-trace on . Then is linear if, and only if, the restriction of to the closed unit ball of is uniformly weakly continuous.

     

Singular integrals with exponential weights

We study the operators

where is the Hardy-Littlewood maximal function, the Hilbert transform or Carleson operator.

Under suitable conditions on the weight of exponential type, we prove boundedness of from spaces, defined on with respect to the measure to with the same density measure. These operators, that arise in questions of harmonic analysis on noncompact symmetric spaces, are bounded from to if and only if .

     

Rational nodal curves with no smooth Weierstrass points

Let denote the rational curve with nodes obtained from the Riemann sphere by identifying 0 with and with for , where is a primitive th root of unity. We show that if is even, then has no smooth Weierstrass points, while if is odd, then has smooth Weierstrass points.

     

Properties that characterize Gaussian periods and cyclotomic numbers

Let be a prime number, a -th primitive root of 1 and the periods of degree of . Write with . Several characterizations of the numbers and (or, equivalently, of the cyclotomic numbers of order ) are given in terms of systems of equations they satisfy and a condition on the linear independence, over , of the or on the irreducibility, over , of the characteristic polynomial of the matrix .