We show that a comeager Π₁¹ hereditary family of compact sets must have a dense subfamily which is also hereditary. Using this, we prove an “abstract” result which implies the existence of independent ℳ ₀-sets, the meagerness of ₀-sets with the property of Baire, and generalizations of some classical results of Mycielski. Finally, we also give some natural examples of true sets.
We initiate the study of sets of p-multiplicity in locally compact groups and their operator versions. We show that a closed subset E of a second countable locally compact group G is a set of p-multiplicity if and only if is a set of operator p-multiplicity. We exhibit examples of sets of p-multiplicity, establish preservation properties for unions and direct products, and prove a p-version of the Stone-von Neumann Theorem.
It is shown that a Sidon set cannot have an integer cluster point in the Bohr topology.
Let be a compact abelian group and the dual group. It is shown that if is a Sidon set, then the interpolating measures on can be obtained as mean of Riesz products. If is a Sidon set tending to infinity, is of first type. Our approach yields in fact elementary proofs of certain characterizations of Sidonicity obtained in G. Pisier, C.R.A.S., Paris Ser. A, 286 (1978), 1003–1006 – Math. Anal. and Appl., Part B, Advances in Math., Suppl. Sts. vol. 7, 685-726 – preprint, using random Fourier...
On étend au cadre des groupes abéliens localement compacts certains résultats obtenus notamment par G. Debs, R. Kaufman, A. Kechris, A. Louveau et J. Saint Raymond sur la structure des fermés d’unicité et d’unicité au sens large du cercle unité. On montre également que de très nombreuses familles de compacts issues de l’Analyse Harmonique sont exactement de troisième classe dans la hiérarchie de Baire. Comme application, on donne une démonstration simple de l’existence d’ensembles de Dirichlet qui...
We are interested in Banach space geometry characterizations of quasi-Cohen sets. For example, it turns out that they are exactly the subsets E of the dual of an abelian compact group G such that the canonical injection is a 2-summing operator. This easily yields an extension of a result due to S. Kwapień and A. Pełczyński. We also investigate some properties of translation invariant quotients of L¹ which are isomorphic to subspaces of L¹.
We construct the following: a perfect non Dirichlet set every proper closed subset of which is Kronecker, A weak Kronecker set which is not an set; an independent countable Dirichlet set which is not Kronecker; a collection of -disjoint Kronecker sets whose union is independent but Helson ; A countable collection of disjoint Kronecker sets whose union is closed and independent but not Helson: a perfect independent Dirichlet set which is not Helson.
We produce several situations where some natural subspaces of classical Banach spaces of functions over a compact abelian group contain the space c₀.