Grothendieck ideals of $\ell_\infty$

Abstract

We answer several questions in the literature concerning the Grothendieck property of ideals of the Banach lattice $\ell_\infty$ that contain $c_0$. Any such ideal can be represented as a space $c_{0,\mathcal I}$ for $\mathcal I$ an ideal over the natural numbers. We provide a characterization of when $c_{0,\mathcal I}$ is a Grothendieck space in terms of finitely additive measures over $\mathcal P(\omega)$ and elements of $\mathcal I$. Using this characterization we show that there are analytic ideals $\mathcal I$ such that $c_{0,\mathcal I}$ is Grothendieck. In the opposite direction we show that for any AD family $\mathcal A$, $c_{0,\mathcal I(\mathcal A)}$ and $C(K_\mathcal A)$ are not Grothendieck spaces, and that for most of the Borel ideals present in the literature, $c_{0,\mathcal I}$ is not Grothendieck. In particular, the family of ideals that do not have the Grothendieck property is cofinal in the Rudin-Keisler order, so the Grothendieck property is not downward closed in the Kat\v{e}tov order. Continuing the work in \cite{Sobota-Zuchowski, Zuchowski}, we also provide similar results on the Nikodym property of the Boolean subalgebras of $\mathcal P(\omega)$ generated by the ideal $\mathcal I$.