Envelopes in Banach spaces

TL;DR

This paper introduces the concept of isometric envelopes in Banach spaces, explores their properties, and applies them to classify certain ultrahomogeneous Banach spaces, including Gurarij and Lebesgue spaces, with implications for their symmetry and extension properties.

Contribution

It defines isometric envelopes in Banach spaces and uses this to analyze ultrahomogeneity, characterizing spaces like Hilbert and Lebesgue spaces through their subspace structures.

Findings

Hilbert space uniquely coincides with its envelope among reflexive spaces.

Reflexive $L_p$ spaces are the only rearrangement invariant spaces with all 1-complemented subspaces as envelopes.

Identifies the unique quotient of $L_p$ by a Hilbertian subspace and the related isometry group embedding.

Abstract

We define the notion of isometric envelope of a subspace in a Banach space, and relate it to a) the mean ergodic projection on the space of fixed points of a semigroup of contractions, b) results on Korovkin sets from the 70's, and c) extension properties of linear isometric embeddings. We use this concept to address the recent conjecture that the Gurarij space and the spaces $L_p$, $p \notin 2\mathbb N+4$ are the only separable Approximately Ultrahomogeneous Banach spaces (a certain multidimensional transitivity of the action of the linear isometry group). The similar conjecture for Fra\"iss\'e Banach spaces (a strenghtening of the Approximately Homogeneous Property) is also considered. We characterize the Hilbert space as the only separable reflexive space in which any closed subspace coincides with its envelope. We compute some envelopes in the case of Lebesgue spaces, showing that the reflexive $L_p$-spaces are the only reflexive rearrangement invariant spaces on $[0,1]$ for which all $1$-complemented subspaces are envelopes. We also identify the isometrically unique "full" quotient space of $L_p$ by a Hilbertian subspace, for appropriate values of $p$, as well as the associated topological group embedding of the unitary group into the isometry group of $L_p$.