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}.
Categories: math.FA