Connes' embedding conjecture and sums of hermitian squares

Markus Schweighofer (1) , Igor Klep (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 IMFM

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 II1-factors are considered instead of matrices. Under the presence of Connes' conjecture, we derive degree bounds for the certificates

Keywords: Sum of squares; Connes' embedding conjecture; Quadratic module; Tracial state; von Neumann algebra

Domains: Operator Algebras [math.OA]; Optimization and Control [math.OC]

Markus Schweighofer, Igor Klep. Connes' embedding conjecture and sums of hermitian squares. Advances in Mathematics, 2008, 217 (4), pp.1816-1837. ⟨10.1016/j.aim.2007.09.016⟩. ⟨hal-00372254⟩