A Nichtnegativstellensatz for polynomials in noncommuting variables

Let S⋃{f} be a set of symmetric polynomials in noncommuting variables. If f satisfies a polynomial identity Σ i h i * fh i = 1 + Σ i g i * s i g i for some s i ∈ S ⋃ {1}, then f is obviously nowhere negative semidefinite on the class of tuples of nonzero operators defined by the system of inequalities s ≥ 0 (s ∈ S). We prove the converse under the additional assumption that the quadratic module generated by S is Archimedean.