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

Abstract

If \alpha and \beta are countable ordinals such that \beta \neq 0, denote by \tilde{T}_{\alpha,\beta} 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_\beta. It is shown that the Bourgain $\ell^{1}$-index of \tilde{T}_{\alpha,\beta} is \omega^{\alpha+\beta.\omega}. In particular, if \alpha =\omega^{\alpha_{1}}. m_{1}+...+\omega^{\alpha_{n}}. m_{n} in Cantor normal form and \alpha_{n} is not a limit ordinal, then there exists a Banach space whose \ell^{1}-index is \omega^{\alpha}.