The ${\ell }^1$-index of Tsirelson type spaces

Abstract

If αand βare countable ordinals such that β\neq 0, denote by \tilde{T}_{α,β} the completion of $c_{00}$ with respect to the implicitly defined norm ||x|| = max{||x||_{c_{0}}, 1/2 sup \sum_{i=1}^{j}||E_{i}x||}, where the supremum is taken over all finite subsets E_{1},...,E_{j} of $\mathbb{N}$ such that $E_1<...<E_j$ and {min E_{1},...,min E_{j}} \in S_β. It is shown that the Bourgain ${\ell }^1$-index of \tilde{T}_{α,β} is ω^{α+β.ω}. In particular, if α=ω^{α_{1}}. m_{1}+...+ω^{α_{n}}. m_{n} in Cantor normal form and α_{n} is not a limit ordinal, then there exists a Banach space whose \ell^{1}-index is ω^α.