Complex structures on twisted Hilbert spaces

Abstract

We investigate complex structures on twisted Hilbert spaces, with special attention paid to the Kalton-Peck Z2 space and to the hyperplane problem. We consider (nontrivial) twisted Hilbert spaces generated by centralizers obtained from an interpolation scale of Köthe function spaces. We show there are always complex structures on the Hilbert space that cannot be extended to the twisted Hilbert space. If, however, the scale is formed by rearrangement invariant Köthe function spaces then there are complex structures on it that can be extended to a complex structure of the twisted Hilbert space. Regarding the hyperplane problem we show that no complex structure on ℓ2 can be extended to a complex structure on an hyperplane of Z2 containing it.

Group actions on twisted sums of Banach spaces

“…The G-centralizers are just special examples of compatible quasi-linear maps for which the derivation is 0. The "compatibility" terminology is inspired from [10], where complex structures where analysed. Conditions for the existence of a "compatible" complex structure, i.e.…”

“…Conditions for the existence of a "compatible" complex structure, i.e. of a complex structure u on X extending to a complex structure on X ⊕ Ω Y and inducing the complex structure v on Y , is the main objective of [10]. The existence of such a complex structure corresponds exactly to the group {1, i, −1, −i}, with the actions defined by i → u and i → v on X and Y respectively, being compatible with Ω in our setting; the derivation condition boils down to 0 = d(−1) = ud(i) + d(i)v, an easy fact which was observed directly in [10]…”

“…This situation was studied partially in various contexts, and it is also our motivation to provide a common setting for these different results. Apart from the theory of L ∞ -centralizers on Köthe spaces, one can mention as a very simple example compatible complex structures as defined in [10] (in which case the group is simply the four elements multiplicative group generated by i). Note that these groups are abelian and therefore amenable.…”

“…This essentially follows from the fact ( [10]) that some complex structure on H does not extend to a complex structure on Z 2 , if we know that this complex structure is unitary. Since this last fact is not obvious in the description given in [10] we give a simpler proof still inspired on their work. For Ψ quasilinear and [x i ] a finite sequence of n normalized vectors quasi-linear, denote as in [10]…”

Type, cotype and twisted sums induced by complex interpolation

“…In that direction, [9] suggests that this problem is related to the singularity of the quotient map in the twisted sum (1.1). So another objective is to obtain twisted sums in which the quotient map is strictly singular.…”