Banach spaces of $\mathcal{I}$-convergent sequences

Abstract

We study the space $c_{0,\mathcal{I}}$ of all bounded sequences $(x_n)$ that $\mathcal{I}$-converge to $0$, endowed with the sup norm, where $\mathcal{I}$ is an ideal of subsets of $\mathbb{N}$. We show that two such spaces, $c_{0,\mathcal{I}}$ and $c_{0,\mathcal{J}}$, are isometric exactly when the ideals $\mathcal{I}$ and $\mathcal{J}$ are isomorphic. Additionally, we analyze the connection of the well-known Katětov pre-order ${\le }_K$ on ideals with some properties of the space $c_{0,\mathcal{I}}$. For instance, we show that $\mathcal{I}{\le }_K\mathcal{J}$ exactly when there is a (not necessarily onto) Banach lattice isometry from $c_{0,\mathcal{I}}$ to $c_{0,\mathcal{J}}$, satisfying some additional conditions. We present some lattice-theoretic properties of $c_{0,\mathcal{I}}$, particularly demonstrating that every closed ideal of ${\ell }_{\infty }$ is equal to $c_{0,\mathcal{I}}$ for some ideal $\mathcal{I}$ on $\mathbb{N}$. We also show that certain classical Banach spaces are isometric to $c_{0,\mathcal{I}}$ for some ideal $\mathcal{I}$, such as the spaces ${\ell }_{\infty }(c_0)$ and $c_0({\ell }_{\infty })$. Finally, we provide several examples of ideals for which $c_{0,\mathcal{I}}$ is not a Grothendieck space.