Real Algebraic and Analytic Geometry
Submission: 2016, September 27.
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. .
Mathematics Subject Classification (2010): 16K20, 11E39, 13J30.
Keywords and Phrases: Central simple algebra, involution, formally real field, hermitian form, signature, positivity, sum of hermitian squares.
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