The Procesi-Schacher conjecture and Hilbert's 17th problem for algebras with involution

TL;DR

This paper investigates the Procesi-Schacher conjecture related to positive elements in algebras with involution, providing counterexamples, identifying cases where it holds, and establishing a Positivstellensatz for noncommutative polynomials.

Contribution

It constructs elementary counterexamples to the Procesi-Schacher conjecture and proves a Positivstellensatz for noncommutative polynomials positive on fixed-size matrices.

Findings

Counterexamples to the conjecture are constructed.

Conditions where the conjecture holds are identified.

A Positivstellensatz for noncommutative polynomials is established.

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.