Banakh, T., Garbulinska-Wegrzyn, J. (2019). A universal Banach space with a $K$-unconditional basis. Advances in Operator Theory, 4(3), 574-586. doi: 10.15352/aot.1805-1369

Taras Banakh; Joanna Garbulinska-Wegrzyn. "A universal Banach space with a $K$-unconditional basis". Advances in Operator Theory, 4, 3, 2019, 574-586. doi: 10.15352/aot.1805-1369

Banakh, T., Garbulinska-Wegrzyn, J. (2019). 'A universal Banach space with a $K$-unconditional basis', Advances in Operator Theory, 4(3), pp. 574-586. doi: 10.15352/aot.1805-1369

Banakh, T., Garbulinska-Wegrzyn, J. A universal Banach space with a $K$-unconditional basis. Advances in Operator Theory, 2019; 4(3): 574-586. doi: 10.15352/aot.1805-1369

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

^{1}Ivan Franko National University of Lviv, Ukraine

^{2}Jan Kochanowski University, Poland

Receive Date: 17 May 2018,
Revise Date: 10 December 2018,
Accept Date: 10 December 2018

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.