Examples of \(k\)-iterated spreading models (Q2847827)

Spreading models, introduced by \textit{A. Brunel} and \textit{L. Sucheston} [Math. Syst. Theory 7 (1973), 294--299 (1974; Zbl 0323.46018)], form a well-known tool extensively used for studying the geometry of Banach spaces. Iterated spreading models are defined inductively: a \((k+1)\)-iterated spreading model of a fixed Banach space \(X\), \(k\geq 1\), is a spreading model of a space generated by a \(k\)-iterated model of \(X\), with standard spreading models treated as 1-iterated spreading models. The question of H.~Rosenthal whether \(k\)-iterated spreading models coincide with 1-iterated ones was answered in the negative for \(k=2\) by \textit{B. Beauzamy} and \textit{B. Maurey} [Ark. Mat. 17, 193--198 (1979; Zbl 0477.46018)].NEWLINENEWLINEIn the paper under review, the authors prove that, for any spreading sequence \((e_n)\) generating a uniformly convex space \(E\), there exists a sequence \(\{X_k\}_{k\in\mathbb{N}}\) of uniformly convex Banach spaces with symmetric bases such that each \(X_k\) admits a \(k\)-iterated spreading model equivalent to \((e_n)\), but none of the spaces generated by \(i\)-iterated spreading models of \(X_k\), \(i<k\), contains \(E\) isomorphically. Among the tools used in the construction, there are techniques of renormings, superreflexive Banach spaces with an unconditional basis introduced by \textit{T. Figiel} and \textit{W. B. Johnson} [Compos. Math. 29, 179--190 (1974; Zbl 0301.46013)], and embedding spaces with an unconditional basis into spaces with a symmetric basis, cf. [\textit{W. J. Davis}, Isr. J. Math. 20, 189--191 (1975; Zbl 0311.46012)].