Year: 2018
Type: preprint
Abstract: For a constant $K\geq 1$ let $\mathfrak{B}_K$ be the class of pairs $(X,(\mathbf e_n)_{n\inω})$ consisting of a Banach space $X$ and an unconditional Schauder basis $(\mathbf e_n)_{n\inω}$ for $X$, having the unconditional basic constant $K_u\leq K$. 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 $\big(\mathbb U_K,(\mathbf e_n)_{n\inω}\big)$ which is $\mathfrak{RI}_K$-universal in the sense that each basis preserving isometry $f:Λ\to\mathbb U_K$ defined on a based subspace $Λ$ 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 $f:Λ\to\mathbb U_K$ defined on a based subspace $Λ$ 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 Pełczyński in 1969.
Source: arXiv (Cornell University)
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2950906178