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

If $\alpha$ and $\beta$ are countable ordinals such that $\beta\neq0$, denote by $\overset{_{\sim}}{T}_{\alpha,\beta}$ the completion of $c_{00}$ with respect to the implicitly defined norm

where the supremum is taken over all finite subsets $E_{1},\dots,E_{j}$ of $\mathbb{N}$ such that $E_{1}<\dots<E_{j}$ and $\{\min E_{1},\dots,\min E_{j}\}\in\mbox{$\mathcal{S}$ }_{\beta}$. It is shown that the Bourgain $\ell^{1}$-index of $\overset{_{\sim}}{T}_{\alpha,\beta}$is $\omega ^{\alpha+\beta\cdot\omega}$. In particular, if $\omega_{1}>\alpha =\omega^{\alpha_{1}}\cdot m_{1}+\dots+\omega^{\alpha_{n}}\cdot m_{n}$ in Cantor normal form and $\alpha_{n}$ is not a limit ordinal, then there exists a Banach space whose $\ell^{1}$-index is $\omega^{\alpha}$.