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

Abstract: For a free filter $F$ on $omega$, endow the space $N_F=omegacup{p_F}$, where $p_F otinomega$, with the topology in which every element of $omega$ is isolated whereas all open neighborhoods of $p_F$ are of the form $Acup{p_F}$ for $Ain F$. 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 a space $N_F$ carries a sequence $langlemu_ncolon ninomega angle$ of normalized finitely supported signed measures such that $mu_n(f) o 0$ for every bounded continuous real-valued function $f$ on $N_F$ if and only if there is a density submeasure $varphi$ on $omega$ such that the dual ideal $F^*$ is contained in the exhaustive ideal $mbox{Exh}(varphi)$. Consequently, we get that if $F$ is a free filter on $omega$ contained in the filter dual to a density ideal, 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. We also prove that the set of filters which can be covered by filters dual to density ideals contains a family of $2^omega$ many mutually non-isomorphic $mathbb{F}_sigma$ P-filters and a family of $2^{2^omega}$ many mutually non-isomorphic non-Borel filters.

This page was built for publication: The Josefson--Nissenzweig theorem and filters on $\omega$