Connes' embedding conjecture and sums of hermitian squares
Operator Algebras
2011-04-19 v1 Rings and Algebras
Abstract
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.
Cite
@article{arxiv.math/0607615,
title = {Connes' embedding conjecture and sums of hermitian squares},
author = {Igor Klep and Markus Schweighofer},
journal= {arXiv preprint arXiv:math/0607615},
year = {2011}
}