Twisted sums of C(K)-spaces
by
Jesus M. F. Castillo
Departamento de Matematicas, Universidad de Extremadura
Coauthors: Felix Cabello Sanchez (Universidad de Extremadura), Nigel Kalton (University of MIssouri), David Yost (King Saudi University)

A twisted sum of two quasi-Banach spaces Y and Z is a quasi-Banach space X admitting a subspace isomorphic to Y and such that the corresponding quotient X/Y is isomorphic to Z. A twisted sum of Y and Z is said to be trivial if the subspace Y is complemented in X. Twisted sums of the classical l_p spaces, 0 < p < +\infty, have been studied by Enflo, Kalton, Lindenstrauss, Peck, Pisier, Ribe and Roberts, among others.

Nevertheless, not many things are known about twisted sums of C(K)-spaces. In this conference we inform about some advances on this problem. In particular, we show:

1) Let K be a compact Hausdorff space for which K^\omega is not empty and let Z be a separable Banach space having some spreading model not isomorphic to l₁. Then there exists a nontrivial twisted sum of C(K) and Z.

2) To be isomorphic to a C(K)-space is not a 3-space property.

3) There exists a twisted sum X of C[0, 1] and c₀ such that the quotient map X --> c₀ is strictly singular.

The result 1) connects with earlier estimates of Amir and Baker; as a corollary one obtains the existence of a nontrivial twisted sum of C(\omega^\omega) and c₀, whch is optimal regarding Sobczyk's theorem. The result 2) solves a question implicit in Bessaga and Pelczynski: if being isomorphic to C[0, 1] is a 3-space property. The result 3) yields the existence of a Banach space X whose dual is isomorphic to L₁, while X itself cannot be renormed so that its dual becomes isometric to L₁. The previously known example was due to Bourgain and Delbaen.

(T)

Date received: November 30, 1999