Local Banach-space dichotomies and ergodic spaces
Abstract
We prove a local version of Gowers' Ramsey-type theorem [Ann. Math. 156 (2002)], as well as local versions both of the Banach space first dichotomy (the "unconditional/HI" dichotomy) of Gowers [Ann. 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.
Keywords
Cite
@article{arxiv.2005.06458,
title = {Local Banach-space dichotomies and ergodic spaces},
author = {W. Cuellar Carrera and N. de Rancourt and V. Ferenczi},
journal= {arXiv preprint arXiv:2005.06458},
year = {2021}
}
Comments
69 pages. As accepted in the Journal of the European Mathematical Society