Abstract: We extend and generalize the result of Kalton and Swanson (Z2 is a symplectic Banach space with no Lagrangian subspace) by showing that all higher order Rochgberg spaces R(n) are symplectic Banach spaces with no Lagrangian subspaces. The nontrivial symplectic structure on Rochberg spaces of even order is the one induced by the natural duality; while the nontrivial symplectic structure on Rochberg spaces of odd order requires perturbation with a complex structure. We will also study symplectic structures on general Banach spaces and, motivated by the unexpected appearance of complex structures, we introduce and study almost symplectic structures.

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