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- Title
Local Banach-space dichotomies and ergodic spaces.
- Authors
Carrera, Wilson Cuellar; de Rancourt, Noé; Ferenczi, Valentin
- 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.
- Subjects
BANACH spaces; ERGODIC theory; SUBSPACES (Mathematics); HILBERT space; RAMSEY theory
- Publication
Journal of the European Mathematical Society (EMS Publishing), 2023, Vol 25, Issue 9, p3537
- ISSN
1435-9855
- Publication type
Article
- DOI
10.4171/JEMS/1257