Connes’ Embedding Conjecture and Sums of Hermitian Squares
نویسندگان
چکیده
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.
منابع مشابه
Correction of a Proof in ”connes’ Embedding Conjecture and Sums of Hermitian Squares”
We show that Connes’ embedding conjecture (CEC) is equivalent to a real version of the same (RCEC). Moreover, we show that RCEC is equivalent to a real, purely algebraic statement concerning trace positive polynomials. This purely algebraic reformulation of CEC had previously been given in both a real and a complex version in a paper of the last two authors. The second author discovered a gap i...
متن کاملTrace-positive polynomials, sums of hermitian squares and the tracial moment problem
A polynomial f in non-commuting variables is trace-positive if the trace of f(A) is positive for all tuples A of symmetric matrices of the same size. The investigation of trace-positive polynomials and of the question of when they can be written as a sum of hermitian squares and commutators of polynomials are motivated by their connection to two famous conjectures: The BMV conjecture from stati...
متن کاملSums of Hermitian Squares and the Bmv Conjecture
We show that all the coefficients of the polynomial tr((A+ tB)) ∈ R[t] are nonnegative whenever m ≤ 13 is a nonnegative integer and A and B are positive semidefinite matrices of the same size. This has previously been known only for m ≤ 7. The validity of the statement for arbitrary m has recently been shown to be equivalent to the Bessis-Moussa-Villani conjecture from theoretical physics. In o...
متن کاملTrace-positive polynomials and the quartic tracial moment problem Polynômes avec une trace positive et le problème quartique des moments traciaux
The tracial analog of Hilbert’s classical result on positive binary quartics is presented: a trace-positive bivariate noncommutative polynomial of degree at most four is a sum of hermitian squares and commutators. This is applied via duality to investigate the truncated tracial moment problem: a sequence of real numbers indexed by words of degree four in two noncommuting variables with values i...
متن کاملSums of Hermitian Squares as an Approach to the Bmv Conjecture
Lieb and Seiringer stated in their reformulation of the BessisMoussa-Villani conjecture that all coefficients of the polynomial p(t) = tr[(A+ B)] are nonnegative whenever A and B are any two positive semidefinite matrices of the same size. We will show that for all m ∈ N the coefficient of t in p(t) is nonnegative, using a connection to sums of Hermitian squares of non-commutative polynomials w...
متن کامل