A Banach space with a countable infinite number of complex structures

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.

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Similarities and differences between real and complex Banach spaces: an overview and recent developments

Moslehian

Muñoz-Fernández

Peralta

et al. 2022

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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There are numerous cases of discrepancies between results obtained in the setting of real Banach spaces and those obtained in the complex context. This article is a modern exposition of the subtle differences between key results and theories for complex and real Banach spaces and the corresponding linear operators between them. We deeply discuss some aspects of the complexification of real Banach spaces and give several examples showing how drastically different can be the behavior of real Banach spaces versus their complex counterparts.

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A Banach space with a countable infinite number of complex structures

Abstract: We give examples of real Banach spaces with exactly infinite countably many complex structures and with $\omega_1$ many complex structures

Cited by 7 publications

References 17 publications

There is no largest proper operator ideal

There is no largest proper operator ideal

Complex structures on twisted Hilbert spaces

Similarities and differences between real and complex Banach spaces: an overview and recent developments

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