No abstract

Abstract. We construct a Bourgain-Delbaen L∞-space XKus with structure that is strongly heterogeneous: any bounded operator on XKus is a compact perturbation of a multiple of the identity, whereas the space XKus is saturated with unconditional basic sequences.

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.