Embedding ₂ and J into subspaces of JT and JT^*

In the first part of the paper we show that every closed subspace of JT or JT^* J T ∗ contains ₂ ℓ 2 complemented in JT or JT^* J T ∗ respectively, and JT contains uncomplemented copies of ₂ ℓ 2 . As a result, the predual B B of JT , as well as the spaces JT and JT^* J T ∗ , are subprojective and superprojective. In the second part, we prove that every weakly Cauchy sequence that is not weakly convergent in JT has a subsequence equivalent to the basis of J . Hence, every non-reflexive subspace of JT contains an isomorphic copy of J , and every Schauder basic sequence in JT has a subsequence which is equivalent either to the basis of ₂ ℓ 2 or to the basis of J . Moreover these subspaces may be selected to be complemented in JT .