Grzegorz Plebanek

A note on extensions of asymptotic density

By a density we mean any extension of the asymptotic density to a finitely additive measure defined on all sets of natural numbers. We consider densities associated to ultrafilters on ω and investigate two additivity properties of such densities. In particular, we show that there is a density ν for which L1(ν) is complete.

Compact spaces that result from adequate families of sets

Abstract We consider compact spaces defined by adequate families of sets as well as continuous images of such spaces which are called AD-compact. The class of AD-compact spaces contains all polyadic spaces. We note some general properties of AD-compact spaces. We prove that there are nonpolyadic AD-compact spaces having a strictly positive measure. We also show that some results on Banach spaces C ( K ) valid for a dyadic K may be extended to K being AD-compact.

A weak* separable C(K)* space whose unit ball is not weak* separable

We provide a ZFC example of a compact space K such that C(K)* is w*-separable but its closed unit ball is not w*-separable. All previous examples of such kind had been constructed under CH. We also discuss the measurability of the supremum norm on that C(K) equipped with its weak Baire sigma-algebra.

Representations of ideals in polish groups and in Banach spaces

We investigate ideals of the form {A⊆ω: ∑ n∈A xn is unconditionally convergent} where (xn)n∈ω is a sequence in a Polish group or in a Banach space. If an ideal on ω can be seen in this form for some sequence in X , then we say that it is representable in X . After numerous examples we show the following theorems: (1) An ideal is representable in a Polish Abelian group iff it is an analytic P-ideal. (2) An ideal is representable in a Banach space iff it is a non-pathological analytic P-ideal. We focus on the family of ideals representable in c0. We characterize this property via the defining sequence of measures. We prove that the trace of the null ideal, Farah’s ideal, and Tsirelson ideals are not representable in c0, and that a tall Fσ P-ideal is representable in c0 iff it is a summable ideal. Also, we provide an example of a peculiar ideal which is representable in `1 but not in R. Finally, we summarize some open problems of this topic.

On Corson compacta and embeddings of

We investigate properties of those compact spaces K for which the Banach space C(K) can be isomorphically embedded into a space C(L), where L is Corson compact. We show that in such a case K must be Corson compact provided K has some additional measure—theoretic property. The result is applicable to Rosenthal compacta and several other classes of compact spaces K.

C(K)

We present a combinatorial description of those families P of sets, for which there is a finite measure μ such that inf{μ(P): P e P} 0. This result yields a topological characterization of measure-compactness and Borel measure-compactness. It is also applied to a problem on the existence of regular measure extensions

spaces

Abstract We show that for an arbitrary Banach space E , the free Banach lattice F B L [ E ] generated by E satisfies the σ -bounded chain condition.

Families of sets of positive measure

Abstract We discuss a problem of Ulam: whether every subset of the plane can be obtained by making countably many set operations with generalized rectangles.