@article {
author = {Banakh, Taras and Garbulinska-Wegrzyn, Joanna},
title = {A universal Banach space with a $K$-unconditional basis},
journal = {Advances in Operator Theory},
volume = {4},
number = {3},
pages = {574-586},
year = {2019},
publisher = {Tusi Mathematical Research Group (TMRG)},
issn = {2538-225X},
eissn = {2538-225X},
doi = {10.15352/aot.1805-1369},
abstract = {For a constant $K\geq 1$ 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 Fra\"iss\'e theory, we construct a rational $K$-based Banach space $\big(\mathbb U_K,(\mathbf e_n)_{n\in\omega}\big)$ 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 $\varepsilon$-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 $\varepsilon$-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 Pe\l czy\'nski in 1969.},
keywords = {Banach space,unconditional Schauder basis,isometric embedding,Fra"iss'e limit},
url = {http://www.aot-math.org/article_80091.html},
eprint = {http://www.aot-math.org/article_80091_91eca671b2ab787f5b6d43080502f526.pdf}
}