Ergodic characterization of reflexivity

A not so recent paper of Fonf, Lin and Wojtaszczyk (Journal of Functional Analysis Volume 187, Issue 1, 1 December 2001, Pages 146-162) proved that any ergodic Banach space with a (Schauder) basis is reflexive solving partially an open question of Sucheston posted more that 40 years ago.

A reasonable question is if we can extend this result when we consider ergodic Banach spaces not necessarily having a Schauder basis. The only know example of Banach spaces with no basis is due to Enflo (Acta Mathematica,Volume 130, Number 1 / july de 1973, Pages 309-317) and it is a non familiar space at all (actually it is a hard construction).

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