The ℓ1-indices of Tsirelson type spaces

Abstract

If α and β are countable ordinals such that β ≠ 0, denote by T̃α,β the completion of c00 with respect to the implicitly defined norm ∥x∥ = max{∥x∥Sα, 1/2 sup ∑i=1 i ∥Eix∥}, where the supremum is taken over all finite subsets E1,..., Ej of ℕ such that E1 α = ωα1·m1+...+ωαn ·mn in Cantor normal form and αn is not a limit ordinal, then there exists a Banach space whose ℓ1-index is ωα.