Complex structures on twisted Hilbert spaces

We investigate complex structures on twisted Hilbert spaces, with special attention paid to the Kalton–Peck Z 2 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 ℓ 2 can be extended to a complex structure on a hyperplane of Z 2 containing it.