The ergodicity of Orlicz sequence spaces

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 ${\mathrm{\ell }}_2(\phi )$ 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.