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

TL;DR

This paper develops a theory of positivity for symmetric elements in algebras with involution, providing a noncommutative analogue of Artin's solution to Hilbert's 17th problem and answering a question by Procesi and Schacher.

Contribution

It introduces a new notion of positivity for symmetric elements and characterizes totally positive elements as sums of hermitian squares, extending classical results to a noncommutative setting.

Findings

Characterization of totally positive elements as sums of hermitian squares

Answer to Procesi and Schacher's question on element representation

Extension of Artin's solution to a noncommutative context

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.