On the hereditary proximity to ℓ1

In the first part of the paper we present and discuss concepts of local and asymptotic hereditary proximity to ℓ1. The second part is devoted to a complete separation of the hereditary local proximity to ℓ1 from the asymptotic one. More precisely for every countable ordinal ξ we construct a separable Hereditarily Indecomposable reflexive space Xξ such that every infinite-dimensional subspace of it has Bourgain ℓ1-index greater than ωξ and the space itself has no ℓ1-spreading model.