Embedding $$\ell _2$$ and J into subspaces of JT and $$JT^*$$

Abstract In the first part of the paper we show that every closed subspace of JT or $$JT^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>J</mml:mi> <mml:msup> <mml:mi>T</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:mrow> </mml:math> contains $$\ell _2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> complemented in JT or $$JT^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>J</mml:mi> <mml:msup> <mml:mi>T</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:mrow> </mml:math> respectively, and JT contains uncomplemented copies of $$\ell _2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> . As a result, the predual $$\mathcal {B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>B</mml:mi> </mml:math> of JT , as well as the spaces JT and $$JT^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>J</mml:mi> <mml:msup> <mml:mi>T</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:mrow> </mml:math> , 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 $$\ell _2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> or to the basis of J . Moreover these subspaces may be selected to be complemented in JT .