Abstract
Positivity , sums of squares and positivstellensatze for * - algebras
Konrad Schmüdgen
Universität Leipzig
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.
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.
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