Tusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X4320190701A universal Banach space with a $K$-unconditional basis5745868009110.15352/aot.1805-1369ENTaras BanakhIvan Franko National University of Lviv, UkraineJoanna Garbulinska-WegrzynJan Kochanowski University, PolandJournal Article20180517For a constant $Kgeq 1$ let $mathfrak{B}_K$ be the class of pairs $(X,(mathbf e_n)_{ninomega})$ consisting of a Banach space $X$ and an unconditional Schauder basis $(mathbf e_n)_{ninomega}$ for $X$, having the unconditional basic constant $K_ule K$. 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 $X$.<br /> Using the technique of Fra"iss'e theory, we construct a rational $K$-based Banach space $big(mathbb U_K,(mathbf e_n)_{ninomega}big)$ which is $mathfrak{RI}_K$-universal in the sense that each basis preserving isometry $f:Lambdatomathbb U_K$ defined on a based subspace $Lambda$ of a finite-dimensional rational $K$-based Banach space $A$ extends to a basis preserving isometry $bar f:Atomathbb U_K$ of the based Banach space $A$.<br /> 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<br /> $varepsilon$-isometry $f:Lambdatomathbb U_K$ defined on a based subspace $Lambda$ of a finite-dimensional $1$-based Banach space $A$ extends to a base preserving $varepsilon$-isometry $bar f:Atomathbb U_K$ of the based Banach space $A$. On the other hand, we show that no almost $mathfrak{FI}_K$-universal based Banach space exists for $K>1$.<br /> 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 Pel czy'nski in 1969.http://www.aot-math.org/article_80091_91eca671b2ab787f5b6d43080502f526.pdf