  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.
