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Transactions of the American Mathematical Society

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A symplectic Banach space with no Lagrangian subspaces

Authors: N. J. Kalton and R. C. Swanson
Journal: Trans. Amer. Math. Soc. 273 (1982), 385-392
MSC: Primary 58B20; Secondary 46B99
MathSciNet review: 664050
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Abstract: In this paper we construct a symplectic Banach space $ (X,\Omega )$ which does not split as a direct sum of closed isotropic subspaces. Thus, the question of whether every symplectic Banach space is isomorphic to one of the canonical form $ Y \times {Y^ \ast }$ is settled in the negative. The proof also shows that $ \mathfrak{L}(X)$ admits a nontrivial continuous homomorphism into $ \mathfrak{L}(H)$ where $ H$ is a Hilbert space.

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Keywords: Symplectic Banach space, isotropic, Lagrangian subspace, strictly singular operators, $ {C^ \ast }$-algebras
Article copyright: © Copyright 1982 American Mathematical Society

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