A Non-Commutative Real Nullstellensatz and Hilbert's 17th Problem

Abstract

1. Generalities Our object of study will be algebras with involution, or *-algebras as we will often call them. All our rings will be algebras over a field F of characteristic # 2; F is endowed with an automorphism a of period one or two with respect to which the involution * will be assumed to be semi-linear. In Section I we identify the central simple algebras which can support an involution of type. In Section IV we prove a Nullstellensatz for representations of prime algebras into such central simple algebras. This is used in Section V to classify the elements of a central simple algebra which are positive in all orderings. Section V follows closely the ideas of Artin-Schreier theory developed in proving that positive-valued rational functions are sums of squares. Let R be a *-algebra. A *-ideal I of R is an ideal which is stable under * R will be called *-simple if it has no non-trivial *-ideals. The *-center of R will be those elements of the center of R which are fixed by *. The *-center of a *-simple algebra is easily seen to be a field. Let R be *-simple with *-center G, G algebraically closed, and R finite dimensional over G. Then by [4, Chapter 0], R is one of the following types: [InJ R G, (n x n matrices over G) with the usual transpose involution; [II,] R -G2 with the usual symplectic involution; [IIIJ R G, ED Go (Go = the opposite ring of G,) with the exchange involution.