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On a question of Haskell P. Rosenthal concerning a characterization of \(c_0\) and \(\ell_p\). (English) Zbl 1067.46010

M. Zippin showed in [Isr. J. Math. 4, 265–272 (1966; Zbl 0148.11202)] that a normalized basis of a Banach space, all of whose normalized block bases are equivalent to it, is equivalent to the unit vector basis of \(c_0\) or \(\ell_p\) for some \(p\geq 1\). H. P. Rosenthal posed the following question: does the above conclusion remain true if it is only supposed that the normalized basis is such that each of its normalized block bases has a subsequence which is equivalent to the starting basis? Such a basis is called a Rosenthal basis. The Galvin–Prikry theorem implies that every Rosenthal basis is subsymmetric.
The authors give a positive answer to this problem in several cases, under some additional natural assumptions. First, they prove that Rosenthal’s problem has an affirmative answer if uniformity is assumed for the extracted subsequence (i.e., there is a constant \(K\geq 1\) such that every (actually: some kind of) normalized block basis of the Rosenthal basis \((e_n)_n\) has a subsequence which is \(K\)-equivalent to \((e_n)_n\)). The proof uses the deep Krivine theorem [J.-L. Krivine, Ann. Math. (2) 104, 1–29 (1976; Zbl 0329.46008); see also B. Maurey, “Type, cotype and \(K\)-convexity”; in: Johnson, W. B. (ed.) et al., Handbook of the geometry of Banach spaces, Vol. 2 (North-Holland, Amsterdam), 1299–1332 (2003; Zbl 1074.46006), Theorem 3]. They prove then that the answer is positive again in \(X\) if the dual Banach space \(X^\ast\), as well as \(X\), has a Rosenthal basis. That uses their above answer, a recent result of G. Androulakis, E. Odell, T. Schlumprecht and N. Tomczak-Jaegermann [“On the structure of the spreading models of a Banach space”, arXiv:math.FA/0305082], and the following lemma, which is easy, via James’s theorem on spaces with unconditional bases, but fundamental:
Lemma 3. If \((e_n)_n\) is a Rosenthal basis in \(X\), then either \((e_n)_n\) is equivalent to the unit vector basis of \(c_0\) or \(\ell_1\), or \(X\) is reflexive; in this latter case, all the spreading models in \(X\) are equivalent to \((e_n)_n\).
Finally, they show that the answer is positive when the choice of the subsequence can be made continuously, from the set of all the normalized block bases of the Rosenthal basis \((e_n)_n\), equipped with the “Ellentuck–Gowers topology”, to the same set equipped with the product of the discrete topology on \(X\).
In the last section of their paper, the authors investigate the case of spreading models. In particular, they notice that their Lemma 3 (stated above) shows that Rosenthal’s problem has an affirmative answer as long as the following one, posed by S. Argyros, has itself a positive answer: Let \(X\) be a Banach space such that all spreading models in \(X\) are equivalent; must these spreading models be equivalent to the unit vector basis of \(c_0\) or \(\ell_p\) for some \(p\geq 1\)?
Reviewer: Daniel Li (Lens)

MSC:

46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46B20 Geometry and structure of normed linear spaces
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