Banach Spaces of the Type of Tsirelson

Abstract . To any pair (M , θ ) , where M is a family of finite subsets of N , compact in the pointwise topology, and 0 < θ < 1 , we associate a Tsirelson-type Banach space T θ M . It is shown that if the Cantor-Bendixson index of M is greater than n and θ > 1 n then T θ M is reflexive. Moreover, if the Cantor-Bendixson index of M is greater than ω then T θ M does not contain any l , while if the Cantor-Bendixson index of M is finite then T θ M contains some l or co . In particular, if M = {A ⊂ N : |A| ≤ n } and 1 n < θ < 1 then T θ M is isomorphic to some l .