Galois extensions, positive involutions and an application to unitary space-time coding

We show that under certain conditions every maximal symmetric subfield of a central division algebra with positive unitary involution $(B,τ)$ will be a Galois extension of the fixed field of $τ$ and will "real split" $(B,τ)$. As an application we show that a sufficient condition for the existence of positive involutions on certain crossed product division algebras, considered by Berhuy in the context of unitary space-time coding, is also necessary, proving that Berhuy's construction is optimal.