In the first part of the paper we show that every closed subspace of JT or <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$JT^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>J</mi> <msup> <mi>T</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> contains <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\ell _2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> complemented in JT or <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$JT^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>J</mi> <msup> <mi>T</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> respectively, and JT contains uncomplemented copies of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\ell _2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>. As a result, the predual <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {B}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">B</mi> </math></EquationSource> </InlineEquation> of JT, as well as the spaces JT and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$JT^*$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>J</mi> <msup> <mi>T</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>, 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 <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\ell _2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> or to the basis of J. Moreover these subspaces may be selected to be complemented in JT.

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

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2026年8月3日

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Spiros A. Argyros,
Manuel González,
Pavlos Motakis

摘要

In the first part of the paper we show that every closed subspace of JT or $JT^*$ $J T^{*}$ contains $\ell _2$ $ℓ_{2}$ complemented in JT or $JT^*$ $J T^{*}$ respectively, and JT contains uncomplemented copies of $\ell _2$ $ℓ_{2}$ . As a result, the predual $\mathcal {B}$ $B$ 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 $\ell _2$ $ℓ_{2}$ or to the basis of J. Moreover these subspaces may be selected to be complemented in JT.

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

摘要