The ergodicity of Orlicz sequence spaces

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 $\ell_2(φ)$ 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.

Theorems & Definitions (36)

• Definition 1.1: Ferenczi--Rosendal 2005
• Conjecture 1.2: Ferenczi--Rosendal 2005
• Theorem 1.3: Cuellar Carrera 2018
• Theorem 1.4
• Theorem 1.5: Anisca
• Theorem 1.7
• Theorem 1.8
• Proposition 2.1
• Theorem 2.2: LindenstraussTzafririI
• Definition 2.3