Local Banach-space dichotomies and ergodic spaces (Q6057871)

Summary: 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.