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

Abstract

<jats:p> In the first part of the paper we show that every closed subspace of <jats:italic>JT</jats:italic> or <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$JT^*$$</jats:tex-math> <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> </jats:alternatives> </jats:inline-formula> contains <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\ell _2$$</jats:tex-math> <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> </jats:alternatives> </jats:inline-formula> complemented in <jats:italic>JT</jats:italic> or <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$JT^*$$</jats:tex-math> <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> </jats:alternatives> </jats:inline-formula> respectively, and <jats:italic>JT</jats:italic> contains uncomplemented copies of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\ell _2$$</jats:tex-math> <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> </jats:alternatives> </jats:inline-formula> . As a result, the predual <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\mathcal {B}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>B</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> of <jats:italic>JT</jats:italic> , as well as the spaces <jats:italic>JT</jats:italic> and <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$JT^*$$</jats:tex-math> <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> </jats:alternatives> </jats:inline-formula> , are subprojective and superprojective. In the second part, we prove that every weakly Cauchy sequence that is not weakly convergent in <jats:italic>JT</jats:italic> has a subsequence equivalent to the basis of <jats:italic>J</jats:italic> . Hence, every non-reflexive subspace of <jats:italic>JT</jats:italic> contains an isomorphic copy of <jats:italic>J</jats:italic> , and every Schauder basic sequence in <jats:italic>JT</jats:italic> has a subsequence which is equivalent either to the basis of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\ell _2$$</jats:tex-math> <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> </jats:alternatives> </jats:inline-formula> or to the basis of <jats:italic>J</jats:italic> . Moreover these subspaces may be selected to be complemented in <jats:italic>JT</jats:italic> . </jats:p>