2026/07/27 by Michael Hrusák, Michael A. Rincón-Villamizar, Luis Sáenz +1
#math.FA
We answer several questions in the literature concerning the Grothendieck property of ideals of the Banach lattice ℓ_∞ that contain c0. Any such ideal can be represented as a space c0,\mathcal I for \mathcal I an ideal over the natural numbers. We provide a characterization of when c0,\mathcal I is a Grothendieck space in terms of finitely additive measures over \mathcal P(ω) and elements of \mathcal I. Using this characterization we show that there are analytic ideals \mathcal I such that c0,\mathcal I is Grothendieck. In the opposite direction we show that for any AD family \mathcal A, c0,\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, c0,\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 \citeSobota-Zuchowski, Zuchowski, we also provide similar results on the Nikodym property of the Boolean subalgebras of \mathcal P(ω) generated by the ideal \mathcal I.