2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/70969We prove a non-commutative version of the Hilbert's 17th problem, giving a characterization of the class of non-commutative polynomials in n-undeterminates that have positive trace when evaluated in n-selfadjoint elements in arbitrary II1 von Neumann algebra. As a corollary we prove that Connes's embedding conjecture is equivalent to a statement that can be formulated entirely in the context of finite matrices.Some minor misprints were correctedOperator Algebras46L10text