Three-space problems for the approximation property

A. Szankowski

doi:10.4171/JEMS/150

Three-space problems for the approximation property

A. Szankowski^*

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

It is shown that there is a subspace Z_q of l_q for 1 < q < 2 which is isomorphic to l_q and such that l _q=Z_q does not have the approximation property (AP). On the other hand, for 2 < p < ∞ there is a subspace Y_p of l_p such that Y_p does not have AP but l_p=Y _p is isomorphic to l_p. The result is obtained by defining random "Enflo-Davie spaces" Y_p which with full probability fail to have AP for all 2 < p ≤ ∞ and have AP for all 1 ≤ p ≤ 2. For 1 < p ≤ 2, Y_p is isomorphic to l_p.

Original language	English
Pages (from-to)	273-282
Number of pages	10
Journal	Journal of the European Mathematical Society
Volume	11
Issue number	2
DOIs	https://doi.org/10.4171/JEMS/150
State	Published - 2009

Keywords

Approximation property
Quotients of banach spaces

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10.4171/JEMS/150

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title = "Three-space problems for the approximation property",

abstract = "It is shown that there is a subspace Zq of lq for 1 < q < 2 which is isomorphic to lq and such that l q=Zq does not have the approximation property (AP). On the other hand, for 2 < p < ∞ there is a subspace Yp of lp such that Yp does not have AP but lp=Y p is isomorphic to lp. The result is obtained by defining random {"}Enflo-Davie spaces{"} Yp which with full probability fail to have AP for all 2 < p ≤ ∞ and have AP for all 1 ≤ p ≤ 2. For 1 < p ≤ 2, Yp is isomorphic to lp.",

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author = "A. Szankowski",

year = "2009",

doi = "10.4171/JEMS/150",

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N2 - It is shown that there is a subspace Zq of lq for 1 < q < 2 which is isomorphic to lq and such that l q=Zq does not have the approximation property (AP). On the other hand, for 2 < p < ∞ there is a subspace Yp of lp such that Yp does not have AP but lp=Y p is isomorphic to lp. The result is obtained by defining random "Enflo-Davie spaces" Yp which with full probability fail to have AP for all 2 < p ≤ ∞ and have AP for all 1 ≤ p ≤ 2. For 1 < p ≤ 2, Yp is isomorphic to lp.

AB - It is shown that there is a subspace Zq of lq for 1 < q < 2 which is isomorphic to lq and such that l q=Zq does not have the approximation property (AP). On the other hand, for 2 < p < ∞ there is a subspace Yp of lp such that Yp does not have AP but lp=Y p is isomorphic to lp. The result is obtained by defining random "Enflo-Davie spaces" Yp which with full probability fail to have AP for all 2 < p ≤ ∞ and have AP for all 1 ≤ p ≤ 2. For 1 < p ≤ 2, Yp is isomorphic to lp.

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KW - Quotients of banach spaces

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JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

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Three-space problems for the approximation property

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