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. For any non-trivial twisted Hilbert space, we show there are always complex structures on the natural copy of the Hilbert space that cannot be extended to the whole space. Regarding the hyperplane problem we show that no complex structure on ℓ2 can be extended to a complex structure on a hyperplane of Z2 containing it.