The Josefson–Nissenzweig theorem and filters on

Abstract For a free filter F on ω, endow the space NF= \pF\ N F = ω ∪ p F, where pF p F ∉ ω, with the topology in which every element of ω is isolated whereas all open neighborhoods of pF p F are of the form A \pF\ A ∪ p F for A F A ∈ F. Spaces of the form NF N F 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, for a filter F, the space NF N F carries a sequence ₙ: n ⟨ μ n: n ∈ ω ⟩ of normalized finitely supported signed measures such that ₙ (f) 0 μ n (f) → 0 for every bounded continuous real-valued function f on NF N F if and only if F^* K {Z} F ∗ ≤ K Z, that is, the dual ideal F^* F ∗ is Katětov below the asymptotic density ideal {Z} Z. Consequently, we get that if F^* K {Z} F ∗ ≤ K Z, then: (1) if X is a Tychonoff space and NF N F is homeomorphic to a subspace of X, then the space Cₚ^* (X) C p ∗ (X) of bounded continuous real-valued functions on X contains a complemented copy of the space c₀ c 0 endowed with the pointwise topology, (2) if K is a compact Hausdorff space and NF N F 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.