## 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

### Citations:

Zbl 0787.46009; Zbl 0692.03031; Zbl 0297.46013
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