Concerning Bourgain's l1-index of a Banach space

A well known argument of James yields that if a Banach space X contains l(1)(n)'s uniformly, then X contains l(1)(n)'s almost isometrically. In the first half of the paper we extend this idea to the ordinal l(1)-indices of Bourgain. In the second half we use our results to calculate the l(1)-index of certain Banach spaces. Furthermore we show that the l(1)-index of a separable Banach space not containing l(1) must be of the form omega(alpha) for some countable ordinal alpha.