Grothendieck and ${\ell }_{\infty }$-Grothendieck subspaces of ${\ell }_{\infty }$

Abstract

Let $\mathfrak{c}=2^{{\aleph }_0}$. We prove that ${\ell }_{\infty }$ contains $2^{\mathfrak{c}}$ pairwise non-isomorphic non-reflexive Grothendieck subspaces. They may all be chosen to contain the canonical copy of $c_0$ and to have infinite-dimensional reflexive quotient. This is optimal and answers by the authors in 2021. We also show that the relative ${\ell }_{\infty }$-Grothendieck property depends on the specified copy of $c_0$ and is not an isomorphic invariant of the subspace. More generally, for every closed $E\subseteq {\ell }_{\infty }$ there is an ${\ell }_{\infty }$-Grothendieck subspace isomorphic to ${\ell }_{\infty }{\oplus }_{\infty }E$; taking $E=c_0$ answers a problem by González et al.

arXiv comment9 pp., accepted for publication in Mathematische Nachrichten