A universal Banach space with a $K$-unconditional basis

Abstract

For a constant $K\ge 1$ let ${\mathfrak{B}}_K$ be the class of pairs $(X,({\mathbf{e}}_n{)}_{n\in \omega })$ consisting of a Banach space $X$ and an unconditional Schauder basis $({\mathbf{e}}_n{)}_{n\in \omega }$ for $X$, having the unconditional basic constant $K_u\le K$. Such pairs are called $K$-based Banach spaces. A based Banach space $X$ is rational if the unit ball of any finite-dimensional subspace spanned by finitely many basic vectors is a polyhedron whose vertices have rational coordinates in the Schauder basis of $X$. Using the technique of Fraïssé theory, we construct a rational $K$-based Banach space $({\mathbb{U}}_K,({\mathbf{e}}_n{)}_{n\in \omega })$ which is ${\mathfrak{RI}}_K$-universal in the sense that each basis preserving isometry $f:\Lambda \to {\mathbb{U}}_K$ defined on a based subspace $\Lambda$ of a finite-dimensional rational $K$-based Banach space $A$ extends to a basis preserving isometry $\bar{f}:A\to {\mathbb{U}}_K$ of the based Banach space $A$. We also prove that the $K$-based Banach space ${\mathbb{U}}_K$ is almost ${\mathfrak{FI}}_1$-universal in the sense that any base preserving $\epsilon$-isometry $f:\Lambda \to {\mathbb{U}}_K$ defined on a based subspace $\Lambda$ of a finite-dimensional $1$-based Banach space $A$ extends to a base preserving $\epsilon$-isometry $\bar{f}:A\to {\mathbb{U}}_K$ of the based Banach space $A$. On the other hand, we show that no almost ${\mathfrak{FI}}_K$-universal based Banach space exists for $K>1$. The Banach space ${\mathbb{U}}_K$ is isomorphic to the complementably universal Banach space for the class of Banach spaces with an unconditional Schauder basis, constructed by Pełczyński in 1969.

arXiv admin note: substantial text overlap with arXiv:1801.07433

References in corpus (1)

• A proof of uniqueness of the Gurarii space