a sum of hermitian squares of *polynomials*, variants of Hilbert’s 17th problem in a *noncommutative* setting have become a topic of current interest with wide-ranging applications (e.g. in control theory, optimization, engineering, mathematical physics, etc.); see [@dOHMP] for a nice survey. Most of these results have a functional analytic flavour and are what Helton et al. call *dimensionfree*, that is, they deal with evaluations of noncommutative polynomials in matrix algebras of arbitrarily large size.

Procesi and Schacher in their 1976 Annals of Mathematics paper [@PS] introduce a notion of orderings on central simple algebras with involution, prove a real Nullstellensatz, and a weak noncommutative version of Hilbert’s 17th problem. A strengthening of the latter is proposed as a conjecture [@PS p. 404]: *In a central simple algebra with involution, a totally positive element is always a sum of hermitian squares.*

We explain in Section \[sec5\] how these results can be applied to study non-dimensionfree positivity of noncommutative polynomials. Roughly speaking, a noncommutative polynomial all of whose evaluations in $n\times n$ matrices (for *fixed* $n$) are positive semidefinite, is a sum of hermitian squares with denominators and weights.

A brief outline of the rest of the paper is as follows: in 