The Josefson--Nissenzweig theorem and filters on $\omega$

For a free filter $F$ on $\omega$, endow the space $N_F=\omega\cup\{p_F\}$, where $p_F\not\in\omega$, with the topology in which every element of $\omega$ is isolated whereas all open neighborhoods of $p_F$ are of the form $A\cup\{p_F\}$ for $A\in F$. 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 $\langle\mu_n\colon n\in\omega\rangle$ of normalized finitely supported signed measures such that $\mu_n(f)\to 0$ for every bounded continuous real-valued function $f$ on $N_F$ if and only if $F^*\le_K\mathcal{Z}$, that is, the dual ideal $F^*$ is Kat\v{e}tov below the asymptotic density ideal $\mathcal{Z}$. Consequently, we get that if $F^*\le_K\mathcal{Z}$, then: (1) if $X$ is a Tychonoff space and $N_F$ 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 $c_0$ 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.