The Josefson--Nissenzweig theorem and filters on $ω$

Abstract

For a free filter F on ω, endow the space NF = ω ∪ {pF }, where pF ∈ ω, with the topology in which every element of ω is isolated whereas all open neighborhoods of pF are of the form A ∪ {pF } for A ∈ F . Spaces of the form NF constitute the class of the simplest non-discrete Tychonoff spaces. The aim of this paper is to study them in the context of the celebrated Josefson-Nissenzweig theorem from Banach space theory. We prove, e.g., that a space NF carries a sequence µn : n ∈ ω of normalized finitely supported signed measures such that µn(f ) → 0 for every bounded continuous real-valued function f on NF if and only if there is a density submeasure ϕ on ω such that the dual ideal F * is contained in the exhaustive ideal Exh(ϕ). Consequently, we get that if F is a free filter on ω contained in the filter dual to a density ideal, then: (1) if X is a Tychonoff space and NF is homeomorphic to a subspace of X, then the space C * p (X) of bounded continuous real-valued functions on X contains a complemented copy of the space c0 endowed with the pointwise topology, (2) if K is a compact Hausdorff space and NF is homeomorphic to a subspace of K, then the Banach space C(K) of continuous real-valued functions on K is not a Grothendieck space. The latter result generalizes the well-known fact stating that if a compact Hausdorff space K contains a non-trivial convergent sequence, then the space C(K) is not Grothendieck. We also prove that the set of filters which can be covered by filters dual to density ideals contains a family of 2 ω many mutually non-isomorphic Fσ P-filters and a family of 2 2 ω many mutually non-isomorphic non-Borel filters.

Grothendieck $C(K)$-spaces and the Josefson--Nissenzweig theorem

“…the space ℓ ∞ or more generally spaces C(K) for extremely disconnected compact spaces K ( [22]), the space H ∞ of bounded analytic functions on the unit disc ( [7]), reflexive spaces, von Neumann algebras ( [36]), injective spaces, spaces C(K) not containing complemented copies of c 0 ( [9]), etc. On the other hand, if a compact space K contains a non-trivial convergent sequence, then the Banach space C(K) does not have the Grothendieck property (see [33] for more general results). A full internal topological characterization of those compact spaces K for which the spaces C(K) are Grothendieck is however unknown (cf.…”