Noncommutative inequalities

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2. Pure trace polynomials as sums of squares of rationals

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Problem 2.1.
[Jurij Volčič] Can a noncommutative polynomial $p$ such that $\mathrm{tr}(p(X)) \geq 0$ for all matrix evaluations be written as a sum of squares of noncommutative rational function plus commutators of noncommutative rationals?
1. Remark. This problem falls between the purely noncommutative setting and the commutative setting: the commutative setting is Hilbert’s 17th problem while in the purely noncommutative setting, polynomials suffice.

Cite this as: AimPL: Noncommutative inequalities, available at http://aimpl.org/noncommineqV.