Complex interpolation and twisted twisted Hilbert spaces

Félix Cabello Sánchez, Jesús Castillo, Nigel Kalton
2015 Pacific Journal of Mathematics  
We show that Rochberg's generalizared interpolation spaces Z^(n) arising from analytic families of Banach spaces form exact sequences 0→ Z^(n)→ Z^(n+k)→ Z^(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 ℓ_p spaces, then Z^(2) becomes the
more » ... -known Kalton-Peck Z_2 space; we then show that Z^(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 ℓ_2 complemented unless n=1. We construct another nontrivial twisted sum of Z_2 with itself that contains ℓ_2 complemented.
doi:10.2140/pjm.2015.276.287 fatcat:zbiyfom7nfdgvguxqsgqy4tqaq