A noncommutative (nc) polynomial is called (globally) trace-positive if its
evaluation at any tuple of operators in a tracial von Neumann algebra has
nonnegative trace. Such polynomials emerge as trace inequalities in several
matrix or operator variables, and are widespread in mathematics and physics.
This paper delivers the first Positivstellensatz for global trace positivity of
nc polynomials. Analogously to Hilbert’s 17th problem in real algebraic
geometry, trace-positive nc polynomials are shown to be weakly sums of
hermitian squares and commutators of regular nc rational functions. In two
variables, this result is strengthened further using a new sum-of-squares
certificate with concrete univariate denominators for nonnegative bivariate
polynomials.
The trace positivity certificates in this paper are obtained by convex
duality through solving the so-called unbounded tracial moment problem, which
arises from noncommutative integration theory and free probability. Given a
linear functional on nc polynomials, the tracial moment problem asks whether it
is a joint distribution of integral operators affiliated with a tracial von
Neumann algebra. A counterpart to Haviland’s theorem on solvability of the
tracial moment problem is established. Moreover, a variant of Carleman’s
condition is shown to guarantee the existence of a solution to the tracial
moment problem. Together with semidefinite optimization, this is then used to
prove that every trace-positive nc polynomial admits an explicit approximation
in the 1-norm on its coefficients by sums of hermitian squares and commutators
of nc polynomials.
