Vol. 276, No. 2, 2015

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ISSN: 0030-8730
Complex interpolation and twisted twisted Hilbert spaces

Félix Cabello Sánchez, Jesús M. F. Castillo and Nigel J. Kalton

Vol. 276 (2015), No. 2, 287–307
Abstract

We show that Rochberg’s generalized interpolation spaces X(n) arising from analytic families of Banach spaces form exact sequences 0 X(n) X(n+k) X(k) 0. We study some structural properties of those sequences; in particular, we show that nontriviality, having strictly singular quotient map, or having strictly cosingular embedding depend only on the basic case n = k = 1. If we focus on the case of Hilbert spaces obtained from the interpolation scale of p spaces, then X(2) becomes the well-known Kalton–Peck space Z2; we then show that X(n) is (or embeds in, or is a quotient of) a twisted Hilbert space only if n = 1,2, which solves a problem posed by David Yost; and that it does not contain 2 complemented unless n = 1. We construct another nontrivial twisted sum of Z2 with itself that contains 2 complemented.

Keywords
Complex interpolation, twisted sums of Banach spaces
Mathematical Subject Classification 2010
Primary: 46B20, 46B70, 46M18, 46M35
Milestones
Received: 1 October 2013
Accepted: 22 August 2014
Published: 15 July 2015
Authors
Félix Cabello Sánchez
Departamento de Matemáticas
Universidad de Extremadura
Avenida de Elvas
06011 Badajoz
Spain
Jesús M. F. Castillo
Departamento de Matemáticas
Universidad de Extremadura
Avenida de Elvas
06011 Badajoz
Spain
Nigel J. Kalton
Department of Mathematics
University of Missouri
Columbia, MO 65211
United States