  We show that Connes' embedding conjecture on von Neumann algebras is
equivalent to the existence of certain algebraic certificates for a polynomial
in noncommuting variables to satisfy the following nonnegativity condition: The
trace is nonnegative whenever self-adjoint contraction matrices of the same
size are substituted for the variables. These algebraic certificates involve
sums of hermitian squares and commutators. We prove that they always exist for
a similar nonnegativity condition where elements of separable II_1-factors are
considered instead of matrices. Under the presence of Connes' conjecture, we
derive degree bounds for the certificates.
