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On strongly asymptotic \(\ell_{p}\) spaces and minimality. (English) Zbl 1139.46013

First of all, recall that an infinite-dimensional Banach space \(X\) is minimal (resp., complementably minimal) if every infinite-dimensional subspace has a subspace (resp., complemented subspace) isomorphic to \(X\). Moreover, for a fixed \(1 \leq p \leq \infty\), a Banach space \(X\) with a basis \((e_i)\) is asymptotic \(l_p\) if there exists \(C < \infty\) and an increasing function \(f: \mathbb N \to \mathbb N\) such that, for all \(n \in \mathbb N\), every normalized block basis \((x_i)_{i=1}^{n}\) of \((e_i)_{i=f(n)}^{\infty}\) is C-equivalent to the unit vector basis of \(l_{p}^{n}\).
As the authors themselves mention in the introduction of the paper under review, their main results can be summarized in the following slightly weaker form: Let \(1 \leq p \leq \infty\) and let \(X\) be a Banach space with a semi-normalized strongly asymptotic \(l_p\) basis \((e_i)\); then:
(a) if \(X\) is minimal and \(1 \leq p <2\), then \(X\) is isomorphic to a subspace of \(l_p\),
(b) if \(X\) is minimal and \(2 \leq p < \infty\), or if \(X\) is complementably minimal and \(1 \leq p \leq \infty\), then \((e_i)\) is equivalent to the unit vector basis of \(l_p\) (or \(c_0\) if \(p= \infty\)).
They present some consequences about the number of non-isomorphic subspaces of a Banach space. Furthermore, for \(1\leq p <2\), they give examples of strongly asymptotic \(l_p\) basic sequences in \(l_p\) spanning minimal spaces that are not isomorphic to \(l_p\). Finally, they also pose some questions.

MSC:

46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46B03 Isomorphic theory (including renorming) of Banach spaces
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