Local Banach-space dichotomies and ergodic spaces
Wilson Cuellar Carrera
Universidade de São Paulo, BrazilNoé de Rancourt
Charles University, Praha, CzechiaValentin Ferenczi
Universidade de São Paulo, Brazil; Sorbonne Université, Paris, France
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Abstract
We prove a local version of Gowers’ Ramsey-type theorem [Ann. of Math. 156 (2002)], as well as local versions both of the Banach space first dichotomy (the “unconditional/HI” dichotomy) of Gowers [Ann. of Math. 156 (2002)] and of the third dichotomy (the “minimal/tight” dichotomy) due to Ferenczi–Rosendal [J. Funct. Anal. 257 (2009)]. 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.
Cite this article
Wilson Cuellar Carrera, Noé de Rancourt, Valentin Ferenczi, Local Banach-space dichotomies and ergodic spaces. J. Eur. Math. Soc. 25 (2023), no. 9, pp. 3537–3598
DOI 10.4171/JEMS/1257