On the spaces dual to combinatorial Banach spaces
Piotr Borodulin-Nadzieja, Sebastian Jachimek, Anna Pelczar-Barwacz
TL;DR
The paper develops a dual-like framework for combinatorial Banach spaces by introducing the quasi-norm $\\|x\\|^{\\mathcal{F}}$, yielding quasi-Banach spaces $X^{\\mathcal{F}}$ that closely mirror duals of $X_{\\mathcal{F}}$. It proves that, for compact hereditary $\\mathcal{F}$, the Banach envelope of $X^{\\mathcal{F}}$ equals $X^*_{\\mathcal{F}}$ and that $(X^{\\mathcal{F}})^* \cong X_{\\mathcal{F}}^{**} \\cong FIN(\\|\cdot\|_{\\mathcal{F}})$, with an embedding $T_{0}$ of norm 1 from $X^{\\mathcal{F}}$ into $X^*_{\\mathcal{F}}$ mapping extreme points accordingly. The authors then prove that when $\\mathcal{F}$ is large (notably Schreier families), $X^{\\mathcal{F}}$ is $\\ell_1$-saturated and does not have the Schur property, offering a simpler proof strategy and broad applicability to Schreier spaces of all orders. These results illuminate the geometry of dual combinatorial spaces and provide tools for analyzing their preduals and envelopes. Overall, the work ties quasi-Banach duals to classical Banach envelopes, expands knowledge about $\1$-saturation phenomena, and has implications for the structure of Schreier-type spaces.
Abstract
We present quasi-Banach spaces which are closely related to the duals of combinatorial Banach spaces. More precisely, for a compact family $\mathcal{F}$ of finite subsets of $ω$ we define a quasi-norm $\lVert \cdot \rVert^\mathcal{F}$ whose Banach envelope is the dual norm for the combinatorial space generated by $\mathcal{F}$. Such quasi-norms seem to be much easier to handle than the dual norms and yet the quasi-Banach spaces induced by them share many properties with the dual spaces. We show that the quasi-Banach spaces induced by large families (in the sense of Lopez-Abad and Todorcevic) are $\ell_1$-saturated and do not have the Schur property. In particular, this holds for the Schreier families.