The ergodicity of Orlicz sequence spaces
No\'e de Rancourt, Ond\v{r}ej Kurka

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 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 constructed by Kalton and Peck are either Hilbertian, or ergodic. This applies in particular to the Kalton--Peck space and all twisted Hilbert spaces generated by complex interpolation between Orlicz sequence spaces.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
