Let A be a (real or complex) * -algebra with involution a -> a*. There are various notions of
positivity of symmetric elements of A. Positive elements can be defined in algebraic terms (for
instance, as finite sums of squares a * a or weighted sums of squares a * ca) or by Hilbert space
representations of A (as elements which are mapped into positive operators under some set of
representations). Positivstellens¨atze deal with the interplay of these notions.
Artin’s theorem on the solution of Hilbert 17th problem says that each positive polynomials
on Rd is a finite sum of squares of rational functions. In the talk we discuss notions of positivity
and versions of ”natural” generalizations of Artin’s theorem to noncommutative * -algebras.
Some new Positivstellens¨atze are presented for * -algebras of matrices over commutative * -
algebras, for the Weyl algebra and for enveloping algebras of Lie algebras.