Connes' embedding conjecture and sums of hermitian squares

Details Connes' embedding conjecture and sums of hermitian squares Connes' embedding conjecture and sums of hermitian squares

2006-07-25

arxiv.org/abs/math/0607615v1

Adv. Math. 217 (2008), no. 4, 1816-1837

Mathematics

Operator Algebras

Scientific paper

10.1016/j.aim.2007.09.016

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.

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