The Josefson-Nissenzweig theorem and filters on ω

For a free filter F on omega, endow the space N-F = omega boolean OR {p(F)}, where p(F )is not an element of omega, with the topology in which every element of omega is isolated whereas all open neighborhoods of p(F) are of the form A boolean OR{p(F)} for A is an element of F. Spaces of the form 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 N-F carries a sequence <mu(n): n is an element of omega > of normalized finitely supported signed measures such that mu(n)(f) -> 0 for every bounded continuous real-valued function f on N-F if and only if F-& lowast; <= k Z, that is, the dual ideal F-& lowast; is Kat & ecaron;tov below the asymptotic density ideal Z. Consequently, we get that if F-& lowast; <= k Z, then: (1) if X is a Tychonoff space and N-F is homeomorphic to a subspace of X, then the space C-p(& lowast;)(X) of bounded continuous real-valued functions on X contains a complemented copy of the space c(0) endowed with the pointwise topology, (2) if K is a compact Hausdorff space and 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.