Hereditary approximation property.

*(English)*Zbl 1272.46012Let \(X\) be a Banach space. The authors say that \(X\) has the hereditary approximation property (HAP) or is an HAPpy space if all closed subspaces of \(X\) have the approximation property. Hilbert spaces are clearly HAPpy. The first HAPpy spaces which are not isomorphic to Hilbert spaces were constructed by W. B. Johnson [Functional analysis, numerical analysis and optimization, Spec. Top. appl. Math., Proc. Semin. GMD, Bonn 1979, 15–26 (1980; Zbl 0442.46011)].

Denote by \[ d_n(X)=\sup \{ d (E, \ell^n_2): E\subset X, ~\dim E=n\} \] the isomorphism constants of \(X\) to \(\ell^n_2\) from \(n\)-dimensional subspaces of \(X\). (Here, \(d(E,F)\) is the Banach-Mazur distance, i.e., the infimum of \(\| T\|\;\| T^{-1}\|\) as \(T\) ranges over all isomorphisms from \(E\) onto \(F\).) The basic theorem of the article under review is the technical Theorem 2.1. It shows that if \((d_n(X))_n\) goes to infinity sufficiently slowly, then \(X\) is an HAPpy space.

The HAPpy spaces constructed in [loc. cit.] are asymptotically Hilbertian. In particular, as was noted in [loc. cit.], they cannot have a symmetric basis unless they are isomorphic to \(\ell_2\). A problem raised in [loc. cit.] was whether there exist HAPpy spaces with a symmetric basis but not isomorphic to \(\ell_2\). Relying on their basic theorem, the authors give an affirmative answer to this old problem by constructing an HAPpy Orlicz sequence space that is not isomorphic to \(\ell_2\).

As another application of the basic theorem, the authors show that there exists a separable infinite-dimensional Banach space \(X\) not isomorphic to a Hilbert space which is complementably universal for all closed subspaces of all of its quotients. In particular, every closed subspace of \(X\) is isomorphic to a complemented subspace of \(X\). Recall that, in contrast, by the classical Lindenstrauss-Tzafriri theorem [J. Lindenstrauss and L. Tzafriri, Isr. J. Math. 9, 263–269 (1971; Zbl 0211.16301)], a Banach space is isomorphic to a Hilbert space whenever all its closed subspaces are complemented.

The basic theorem of the present article, as was already mentioned, shows that \(X\) is an HAPpy space whenever \(d_n(X)\to\infty\) sufficiently slowly. An important ingredient in its proof is Lemma 2, the main lemma, on the structure of uniformly convex spaces, characterized in terms of norms of finite-rank operators. The main lemma can be applied thanks to the result of G. Pisier [Isr. J. Math. 20, 326–350 (1975; Zbl 0344.46030)] that if \(d_n(X)\to\infty\) sufficiently slowly, then \(X\) is super-reflexive and therefore, by a classical theorem of P. Enflo [Isr. J. Math. 13, 281–288 (1972; Zbl 0259.46012)], \(X\) admits an equivalent uniformly convex norm.

A long list of open questions concludes this inspiring paper.

Denote by \[ d_n(X)=\sup \{ d (E, \ell^n_2): E\subset X, ~\dim E=n\} \] the isomorphism constants of \(X\) to \(\ell^n_2\) from \(n\)-dimensional subspaces of \(X\). (Here, \(d(E,F)\) is the Banach-Mazur distance, i.e., the infimum of \(\| T\|\;\| T^{-1}\|\) as \(T\) ranges over all isomorphisms from \(E\) onto \(F\).) The basic theorem of the article under review is the technical Theorem 2.1. It shows that if \((d_n(X))_n\) goes to infinity sufficiently slowly, then \(X\) is an HAPpy space.

The HAPpy spaces constructed in [loc. cit.] are asymptotically Hilbertian. In particular, as was noted in [loc. cit.], they cannot have a symmetric basis unless they are isomorphic to \(\ell_2\). A problem raised in [loc. cit.] was whether there exist HAPpy spaces with a symmetric basis but not isomorphic to \(\ell_2\). Relying on their basic theorem, the authors give an affirmative answer to this old problem by constructing an HAPpy Orlicz sequence space that is not isomorphic to \(\ell_2\).

As another application of the basic theorem, the authors show that there exists a separable infinite-dimensional Banach space \(X\) not isomorphic to a Hilbert space which is complementably universal for all closed subspaces of all of its quotients. In particular, every closed subspace of \(X\) is isomorphic to a complemented subspace of \(X\). Recall that, in contrast, by the classical Lindenstrauss-Tzafriri theorem [J. Lindenstrauss and L. Tzafriri, Isr. J. Math. 9, 263–269 (1971; Zbl 0211.16301)], a Banach space is isomorphic to a Hilbert space whenever all its closed subspaces are complemented.

The basic theorem of the present article, as was already mentioned, shows that \(X\) is an HAPpy space whenever \(d_n(X)\to\infty\) sufficiently slowly. An important ingredient in its proof is Lemma 2, the main lemma, on the structure of uniformly convex spaces, characterized in terms of norms of finite-rank operators. The main lemma can be applied thanks to the result of G. Pisier [Isr. J. Math. 20, 326–350 (1975; Zbl 0344.46030)] that if \(d_n(X)\to\infty\) sufficiently slowly, then \(X\) is super-reflexive and therefore, by a classical theorem of P. Enflo [Isr. J. Math. 13, 281–288 (1972; Zbl 0259.46012)], \(X\) admits an equivalent uniformly convex norm.

A long list of open questions concludes this inspiring paper.

Reviewer: Eve Oja (Tartu)

##### MSC:

46B28 | Spaces of operators; tensor products; approximation properties |

46B03 | Isomorphic theory (including renorming) of Banach spaces |

46B07 | Local theory of Banach spaces |

46B25 | Classical Banach spaces in the general theory |

47B10 | Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) |

##### Keywords:

Banach spaces; hereditary uniform approximation property; asymptotically Hilbertian spaces; trace formula; hereditary approximation property
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\textit{W. B. Johnson} and \textit{A. Szankowski}, Ann. Math. (2) 176, No. 3, 1987--2001 (2012; Zbl 1272.46012)

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##### References:

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