We show that under certain conditions every maximal symmetric subfield of a central division algebra with positive unitary involution $ (B, \tau) $ will be a Galois extension of the fixed field of $ \tau $ and will "real split" $ (B, \tau) $. As an application we show that a sufficient condition for the existence of positive involutions on certain crossed product division algebras over number fields, considered by Berhuy in the context of unitary space-time coding, is also necessary, proving that Berhuy's construction is optimal.
Citation: |
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V. Astier and T. Unger, Positive cones on algebras with involution, preprint, arXiv: 1609.06601.
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V. Astier and T. Unger, Signatures of hermitian forms, positivity, and an answer to a question of Procesi and Schacher, J. Algebra, 508 (2018), 339-363.
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G. Berhuy, Algebraic space-time codes based on division algebras with a unitary involution, Adv. Math. Commun., 8 (2014), 167-189.
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