Local Banach-space dichotomies and ergodic spaces

Wilson Cuellar Carrera (1) , Noé de Rancourt (2, 3, 4, 5, 6) , Valentin Ferenczi (1, 3)

1 USP - Universidade de São Paulo = University of São Paulo
2 LPP - Laboratoire Paul Painlevé - UMR 8524
3 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
4 UK - Univerzita Karlova [Praha, Česká republika] = Charles University [Prague, Czech Republic]
5 Universität Wien = University of Vienna
6 DMA - Département de Mathématiques et Applications - ENS Paris

We prove a local version of Gowers' Ramsey-type theorem [25], as well as local versions both of the Banach space first dichotomy (the "unconditional/HI" dichotomy) of Gowers [25] and of the third dichotomy (the "minimal/tight" dichotomy) due to Ferenczi-Rosendal [22]. This means that we obtain versions of these dichotomies restricted to certain families of subspaces called D-families, of which several concrete examples are given. As a main example, non-Hilbertian spaces form D-families; therefore versions of the above properties for non-Hilbertian spaces appear in new Banach space dichotomies. As a consequence we obtain new information on the number of subspaces of non-Hilbertian Banach spaces, making some progress towards the "ergodic" conjecture of Ferenczi-Rosendal and towards a question of Johnson.

Mots clés: Ergodic Banach spaces; Ramsey theory; Banach-space dichotomies; Non-Hilbertian spaces; Minimal Banach spaces; Hereditarily indecomposable Banach spaces

Domaines: Analyse fonctionnelle [math.FA]; Combinatoire [math.CO]; Logique [math.LO]

Wilson Cuellar Carrera, Noé de Rancourt, Valentin Ferenczi. Local Banach-space dichotomies and ergodic spaces. Journal of the European Mathematical Society, 2022, 25 (9), pp.3537-3598. ⟨10.4171/JEMS/1257⟩. ⟨hal-04406829⟩