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Characterizations of spreading models of \(\ell ^1\). (English) Zbl 1039.46010

Let \((f_k)\) be a uniformly bounded sequence of real valued functions on a compact space K. H. Rosenthal [Proc. Natl. Acad. Sci. USA, 71, 2411–2413 (1974; Zbl 0297.46013] proved in 1974 that \((f_k)\) has a subsequence equivalent to the unit basis of \(\ell ^1\) if \((f_k)\) contains no pointwise convergent subsequence. The paper under review presents some local analogues of this result by “spreading” models of \(\ell ^1\). Given an ordinal \(1\leq \xi <\omega _1\), the sequence \((f_k)\) is called an \(\ell ^1_\xi \) spreading model if there are constants \(c>0, C>0\) such that \(c\sum ^m_{i=1}| a_i| \leq \| \sum ^m_{i=1}a_i f_{k_i}\| _\infty \leq C\sum ^m_{i=1}| a_i| \) for every \(F=\{k_1,\ldots ,k_m\}\in {\mathcal F}_\xi \) and for every set of real numbers \(a_1,\ldots ,a_m\), where \({\mathcal F}_\xi \) is the so called generalized Schreier family introduced by D. E. Alspach and S. Argyros [Disser. Math. 321, 1–44 (1992; Zbl 0787.46009)]. (Note that \({\mathcal F}_1\) is the usual Schreier family.) The symbol \(\gamma ((f_k))\) indicates the convergence index introduced by A. S. Kechris and A. Louveau [Trans. Am. Math. Soc. 318, 209–236 (1990; Zbl 0692.03031)].
Theorems. Let \((f_k)\) be pointwise convergent. If \(\gamma \big ((f_{n_k})\big )>\omega ^\xi \) for every strictly increasing sequence \((n_k)\), then there exists a strictly increasing sequence \((m_k)\) such that the sequence \((f_{m_k})\) is an \(\ell ^1_\xi \) spreading model. If \((f_k)\) is an \(\ell ^1_\xi \) spreading model, then \(\gamma \big ((f_{n_k})\big )>\omega ^\xi \) for every strictly increasing sequence \((n_k)\). Some relations between \(\ell ^1_\xi \) spreading models and subfamilies of Baire one functions are also presented.
Note that the case when \(\xi =1\) is a result of R. Haydon, E. Odell and H. Rosenthal. The paper is a continuation of earlier works of the author as well as of his collaborators S. A. Argyros, S. Mercourakis and A.Tsarpalias from Athens. It contains a few fine combinatorial lemmata.

MSC:

46B20 Geometry and structure of normed linear spaces
46E99 Linear function spaces and their duals
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