Non-ergodic Banach spaces are near Hilbert

TL;DR

This paper proves that non-ergodic Banach spaces are close to Hilbert spaces, confirming that only $ ext{ell}_2$ might be non-ergodic, and resolves a long-standing question about universal subspaces.

Contribution

It establishes that non-ergodic Banach spaces are near Hilbert and solves a 1976 open problem about the non-existence of certain universal subspaces.

Findings

Non-ergodic Banach spaces are near Hilbert spaces.

$ ext{ell}_p$ ($2<p< finite$) is ergodic.

No separable Banach space is complementably universal for subspaces of $ ext{ell}_p$ for $1 extless p extless 2$.

Abstract

We prove that a non ergodic Banach space must be near Hilbert. In particular, $\ell_p$ ($2<p<\infty$) is ergodic. This reinforces the conjecture that $\ell_2$ is the only non ergodic Banach space. As an application of our criterion for ergodicity, we prove that there is no separable Banach space which is complementably universal for the class of all subspaces of $\ell_p$, for $1\leq p <2$. This solves a question left open by W. B. Johnson and A. Szankowski in 1976.