Signatures of hermitian forms and the Knebusch Trace Formula

TL;DR

This paper extends the Knebusch Trace Formula to hermitian forms over algebras with involution, resolving sign ambiguities present in previous Morita theory-based approaches.

Contribution

It establishes a hermitian version of the Knebusch Trace Formula, providing a new tool to address sign ambiguities in signatures of hermitian forms.

Findings

Hermitian version of the Knebusch Trace Formula established

Sign ambiguities in signatures resolved using the new formula

Erratum corrects previous definitions and omissions

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. The last page is an erratum for the published version. We inadvertently (I) gave an incorrect definition of adjoint involutions; (II) omitted dealing with the case $(H\times H, \widehat{\phantom{m}}\,)$. As $W(H\times H, \widehat{\phantom{m}}\,)= W(R\times R, \widehat{\phantom{m}}\,)=0$, the omission does not affect our reasoning or our results. For the sake of completeness we point out where some small changes should be made in the published version.