---
abstract: 'In 1976 Procesi and Schacher developed an Artin–Schreier type theory for central simple algebras with involution and conjectured that in such an algebra a totally positive element is always a sum of hermitian squares. In this paper elementary counterexamples to this conjecture are constructed and cases are studied where the conjecture does hold. Also, a Positivstellensatz is established for noncommutative polynomials, positive semidefinite on all tuples of matrices of a fixed size.'
address:
- 'University of California San Diego, Department of Mathematics, 9500 Gilman Drive, La Jolla, California 92093-0112, USA'
- 'School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland'
author:
- Igor Klep
- Thomas Unger
date: 18 October 2008
title: 'The Procesi–Schacher conjecture and Hilbert’s 17th problem for algebras with involution'
---

*Dedicated to David W. Lewis on the occasion of his 65th birthday.*

Introduction
============

Artin’s 1927 affirmative solution of Hilbert’s 17th problem (*Is every nonnegative real polynomial a sum of squares of rational functions?*) arguably sparked the beginning of the field of real algebra and consequently real algebraic geometry (cf. [@BCR; @PD]).

Starting with Helton’s seminal paper [@Hel], in which he proved that every positive semidefinite real or complex noncommutative polynomial is 