The ergodicity of Orlicz sequence spaces

TL;DR

The paper demonstrates that non-Hilbertian separable Orlicz sequence spaces are ergodic, containing many non-isomorphic subspaces, and extends this result to certain twisted Hilbert spaces, revealing their structural complexity.

Contribution

It proves ergodicity of non-Hilbertian Orlicz sequence spaces and shows they contain continuum many non-isomorphic subspaces, extending to certain twisted Hilbert spaces.

Findings

Non-Hilbertian Orlicz sequence spaces are ergodic.

Each such space contains continuum many non-isomorphic subspaces.

Twisted Hilbert spaces are either Hilbertian or ergodic.

Abstract

We prove that non-Hilbertian separable Orlicz sequence spaces are ergodic, i.e., the equivalence relation $\mathbb{E}_0$ Borel reduces to the isomorphism relation between subspaces of every such space. This is done by exhibiting non-Hilbertian asymptotically Hilbertian subspaces in those spaces, and appealing to a result by Anisca. In particular, each non-Hilbertian Orlicz sequence space contains continuum many pairwise non-isomorphic subspaces. As a consequence, we prove that the twisted Hilbert spaces $\ell_2(\phi)$ constructed by Kalton and Peck are either Hilbertian, or ergodic. This applies in particular to the Kalton--Peck space $Z_2$ and all twisted Hilbert spaces generated by complex interpolation between Orlicz sequence spaces.