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.
COBISS.SI-ID: 14569561
Recently Lieb and Seiringer showed that the Bessis-Moussa-Villani conjecture from quantum physics can be restated in the following purely algebraic way: The sum of all words in two positive semidefinite matrices where the number of each of the two letters is fixed is always a matrix with nonnegative trace. We show that this statement holds if the words are of length at most 13. In our proof, we establish a connection to sums of hermitian squares of polynomials in noncommuting variables and to semidefinite programming.
COBISS.SI-ID: 14975321
In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. We use a class of functions (NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. In this paper we characterize NC analytic maps that send dimension-free matrix balls to dimension-free matrix balls and carry the boundary to the boundary. Up to normalization, such a map is the direct sum of the identity with an NC analytic map of the ball into the ball.
In 1976 Procesi and Schacher developed an Artin-Schreier type theory for central simple algebras with involution and conjectured that in such an algebra a totally positive element is always a sum of hermitian squares. In this paper elementary counterexamples to this conjecture are constructed and cases are studied where the conjecture does hold. Also, a Positivstellensatz is established for noncommutative polynomials, positive semidefinite on all tuples of matrices of a fixed size.
An algorithm for nding sums of hermitian squares decompositions for polynomials in noncommuting variables is presented. The algorithm is based on the \Newton chip method", a noncommutative analog of the classical Newton polytope method, and semidenite programming.