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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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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DOI: http://dx.doi.org/10.1090/S0002-9947-1982-0664050-6
PII: S 0002-9947(1982)0664050-6
Keywords: Symplectic Banach space, isotropic, Lagrangian subspace, strictly singular operators, $ {C^ \ast }$-algebras
Article copyright: © Copyright 1982 American Mathematical Society