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Singular twisted sums generated by complex interpolation

Abstract: We present new methods to obtain singular twisted sums X??X (i.e., exact sequences 0 ? X ? X ?? X ? X ? 0 in which the quotient map is strictly singular) when X is an interpolation space arising from a complex interpolation scheme and ? is the induced centralizer. Although our methods are quite general, we are mainly concerned with the choice of X as either a Hilbert space or Ferenczi?s uniformly convex Hereditarily Indecomposable space. In the first case, we construct new singular twisted Hilbert spaces (which includes the only known example so far: the Kalton-Peck space Z2). In the second case we obtain the first example of an H.I. twisted sum of an H.I. space. During our study of singularity we introduce the notion of a disjointly singular twisted sum of K¨othe function spaces and construct several examples involving reflexive p-convex K¨othe function spaces (which includes the function space version of the Kalton-Peck space Z2). We then use Rochberg?s description of iterated twisted sums to show that there is a sequence Fn of H.I. spaces so that Fm+n is a singular twisted sum of Fm and Fn, while for l > n the direct sum Fn ?Fl+m is a nontrivial twisted sum of Fl and Fm+n.

 Fuente: Transactions of the American Mathematical Society (2017) Vol. 369, N.7, pp. 4671-4708

 Editorial: American Mathematical Society

 Año de publicación: 2017

 Nº de páginas: 38

 Tipo de publicación: Artículo de Revista

 DOI: 10.1090/tran/6809

 ISSN: 0002-9947,1088-6850

 Proyecto español: MTM2013-45643