Abstract. Signatures of quadratic forms have been generalized to hermitian forms over algebras with involution. In the literature this is done via Morita theory, which causes sign ambiguities in certain cases. In this paper, a hermitian version of the Knebusch Trace Formula is established and used as a main tool to resolve these ambiguities.

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In this paper a further study is made of H-signatures of hermitian forms, introduced previously by the authors. It is shown that a tuple of reference forms H may be replaced by a single form and that the H-signature is invariant under Morita equivalence of algebras with involution. The "prime ideals" of the Witt group are studied, obtaining results that are analogues of the classification of prime ideals of the Witt ring by Harrison and Lorenz-Leicht. It follows that H-signatures canonically correspond to morphisms into the integers.

We extend the classical links between valuations and orderings on fields to Tignol-Wadsworth gauges and positive cones on algebras with involution. We also study the compatibility of gauges and positive cones and prove a corresponding Baer-Krull theorem.

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