Local Banach-space dichotomies and ergodic spaces

Variants of the James Tree space

“…These two properties show that X eh is an ℓ 2 saturated d 2 -H.I. space, a class of Banach spaces defined recently by W. Cuellar Carrera, N. de Rancourt and V. Ferenczi in [16]. Our aim in the present paper is to present James Tree spaces containing ℓ 2 and satisfying the property that every Hilbertian subspace is complemented.…”

“…Important contributions to the program have been made by V. Ferenczi and C. Rosendal [18], [19], [20]. Recently, in [16], W. Cuellar Carrera, N. de Rancourt and V. Ferenczi initiated a theory of classifying Banach spaces with respect to the properties of their subspaces restricted within a predetermined class. The class of non-Hilbertian spaces is of particular interest.…”

“…The following definition of W. Cuellar Carrera, N. de Rancourt and V. Ferenczi is the foremost motivating concept behind this work. Definition 2 ( [16]). A Banach space is called d 2 -hereditarily indecomposable (or d 2 -HI) if it is non-Hilbertian and it does not contain the direct sum of two non-Hilbertian spaces.…”

“…One of the restricted dichotomy theorems from [16] states that a non-Hilbertian Banach space contains a non-Hilbertian subspace that either has an unconditional finite dimensional decomposition or is d 2 -HI. The list of currently known d 2 -HI spaces is brief.…”

Parametrizing the Ramsey theory of block sequences I: Discrete vector spaces

“…Given a family H, the restricted Gowers game G H [X] played below an X ∈ H is defined exactly as G[X], except that we require I's moves to be from H ↾ X. This game has been extensively studied by Noé de Rancourt and coauthors, see [5] and [6]. It enables a variation of Theorem 1.4 for (p)-families without the extra assumption of being full, at the expense of weakening the conclusion for II from G[X] to G H [X] (Theorem 3.3 in [6]).…”