A positive cone on an algebra with involution determines the kernels of the signature map of hermitian forms, yielding corrected proofs of the classification of positive cones and of their behavior under field extensions.
Pfister's local-global principle for Azumaya algebras with involution
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We prove Pfister's local-global principle for hermitian forms over Azumaya algebras with involution over semilocal rings, and show in particular that the Witt group of nonsingular hermitian forms is $2$-primary torsion. Our proof relies on a hermitian version of Sylvester's law of inertia, which is obtained from an investigation of the connections between a pairing of hermitian forms extensively studied by Garrel, signatures of hermitian forms, and positive semidefinite quadratic forms.
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Signature maps from positive cones on algebras with involution
A positive cone on an algebra with involution determines the kernels of the signature map of hermitian forms, yielding corrected proofs of the classification of positive cones and of their behavior under field extensions.