Real Algebraic and Analytic Geometry |

Connes' embedding conjecture and sums of hermitian squares.

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homepage: http://perso.univ-rennes1.fr/markus.schweighofer/

Submission: 2006, July 24.

*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.There is an erratum available here.*

Mathematics Subject Classification (2000): 11E25, 13J30, 58B34.

Keywords and Phrases: sum of squares, Connes' embedding conjecture, quadratic module, tracial state, von Neumann algebra.

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