Separable elastic Banach spaces are universal

TL;DR

The paper proves that all separable infinite-dimensional elastic Banach spaces are universal for all separable Banach spaces, confirming a conjecture and using advanced embedding techniques.

Contribution

It establishes that separable elastic Banach spaces are universal, confirming Johnson and Odell's conjecture, and introduces a generalized basis index for Banach space embeddings.

Findings

C[0,1] embeds into all separable elastic Banach spaces

Separable elastic Banach spaces are universal for all separable Banach spaces

Develops a generalized Bourgain basis index

Abstract

A Banach space $X$ is elastic if there is a constant $K$ so that whenever a Banach space $Y$ embeds into $X$, then there is an embedding of $Y$ into $X$ with constant $K$. We prove that $C[0,1]$ embeds into separable infinite dimensional elastic Banach spaces, and therefore they are universal for all separable Banach spaces. This confirms a conjecture of Johnson and Odell. The proof uses incremental embeddings into $X$ of $C(K)$ spaces for countable compact $K$ of increasing complexity. To achieve this we develop a generalization of Bourgain's basis index that applies to unconditional sums of Banach spaces and prove a strengthening of the weak injectivity property of these $C(K)$ that is realized on special reproducible bases.