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 Katětov 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 $\mathcal{I}$.