On Uniformly finitely extensible Banach spaces

TL;DR

This paper investigates the properties of Uniformly Finitely Extensible Banach spaces, establishing their approximation and extension properties, and explores their connection to automorphic spaces, especially in the context of Hereditarily Indecomposable and asymptotic $ ext{l}_2$ spaces.

Contribution

It demonstrates that UFO spaces possess the Uniform Approximation Property and are compactly extensible, and characterizes automorphic spaces among UFOs, especially in the HI and asymptotic $ ext{l}_2$ cases.

Findings

UFO spaces have the Uniform Approximation Property.

UFO spaces are compactly extensible.

Spaces with all subspaces UFO are automorphic if HI, or Hilbert spaces under certain conditions.

Abstract

We continue the study of Uniformly Finitely Extensible Banach spaces (in short, UFO) initiated in Moreno-Plichko, \emph{On automorphic Banach spaces}, Israel J. Math. 169 (2009) 29--45 and Castillo-Plichko, \emph{Banach spaces in various positions.} J. Funct. Anal. 259 (2010) 2098-2138. We show that they have the Uniform Approximation Property of Pe\l czy\'nski and Rosenthal and are compactly extensible. We will also consider their connection with the automorphic space problem of Lindenstrauss and Rosenthal --do there exist automorphic spaces other than $c_0(I)$ and $\ell_2(I)$?-- showing that a space all whose subspaces are UFO must be automorphic when it is Hereditarily Indecomposable (HI), and a Hilbert space when it is either locally minimal or isomorphic to its square. We will finally show that most HI --among them, the super-reflexive HI space constructed by Ferenczi-- and asymptotically $\ell_2$ spaces in the literature cannot be automorphic.