On the hereditary proximity to ℓ1

Abstract

In the first part of the paper we present and discuss concepts of local and asymptotic hereditary proximity to 1 . The second part is devoted to a complete separation of the hereditary local proximity to 1 from the asymptotic one. More precisely for every countable ordinal ξ we construct a separable Hereditarily Indecomposable reflexive space X ξ such that every infinite-dimensional subspace of it has Bourgain 1 -index greater than ω ξ and the space itself has no 1 -spreading model.

On spaces admitting no $\ell_p$ or $c_0$ spreading model

“…A variant of the next theorem can be found in [4,Theorem 11.3]. We include the proof here to give a more complete presentation.…”

“…equals ω 1 ) then the space contains ℓ 1 . The main result of [4] states that for each countable ordinal ξ there is a separable space X ξ that does not admit an ℓ 1 spreading model and has hereditary ℓ 1 -index greater than ω ξ . If a space X has, as quotient, every space not admitting an ℓ 1 spreading model it must have have the space X ξ as quotient for each ξ < ω 1 .…”

“…where the above supremum is taken over all families (s i ) d i=1 of pairwise incomparable non-empty segments of N <N . For every well-founded tree S of natural numbers, let J S 2,1 be the closed subspace supported on the coordinates of S. Using arguments similar to those in [4], for every well-founded tree S, the space J S 2,1 has no ℓ 1 spreading model. It is easy to see that the Bourgain ℓ 1 index of J S 2,1 is at least the height of the well founded tree S. Arguing as before, we conclude that any space having each J S 2,1 as a quotient must contain ℓ 1 .…”

“…Gowers' construction [10] of a space not containing c 0 , ℓ 1 or a reflexive subspace. Recently, in [4], it was shown that there exist hereditarily indecomposable spaces not admitting any ℓ p or c 0 as a spreading model. In [1], the authors construct a space not admitting an ℓ p , c 0 or reflexive spreading model.…”

Variants of the James Tree space

“…For the definition of the space X eh a norming set W eh is used. Its construction requires the attractors method from [6] (see also [9]). A necessary ingredient is a ground set G. This ground set defines a James Tree variant that we denote by JT G .…”