The ergodicity dichotomy for separable Banach spaces

A separable Banach space is a Banach space with a countable dense subset, and a Banach space is ergodic when the isomorphism relation between its infinite-dimensional subspaces is as complex as the standard relation $E_0$ of eventual equality on $2^{\mathbb N}$. The ergodicity dichotomy. Every separable Banach space is either isomorphic to $\ell_2$ or ergodic.

The preceding structural results establish ergodicity for several classes of Banach spaces, while the remaining minimal case includes classical spaces such as $c_0$, $\ell_p$, and Schlumprecht's space. The conjecture proposes that the only separable non-ergodic space, up to isomorphism, is the Hilbert space $\ell_2$.