Signatures of hermitian forms, positivity, and an answer to a question of Procesi and Schacher

2015/11/19 by Astier, Vincent, Unger, Thomas · 3 citations
#11E39 #13J30 #16K20 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1511.06330

Abstract

Using the theory of signatures of hermitian forms over algebras with involution, developed by us in earlier work, we introduce a notion of positivity for symmetric elements and prove a noncommutative analogue of Artin's solution to Hilbert's 17th problem, characterizing totally positive elements in terms of weighted sums of hermitian squares. As a consequence we obtain an earlier result of Procesi and Schacher and give a complete answer to their question about representation of elements as sums of hermitian squares.

Cited by

• Continuity of total signature maps for Azumaya algebras with involution
• Maximal prepositive cones on quaternion algebras with involution
• Signature maps from positive cones on algebras with involution

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• A Positivstellensatz on the Matrix Algebra of Finitely Generated Free Group
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• A noncommutative generalization of Hunter's positivity theorem
• An Artin-Schreier-type theory for signatures of hermitian forms over involutions