Connes' embedding conjectures and sums of hermitian squares

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2008
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Advances in Mathematics. 2008, 217(4), pp. 1816-1837. ISSN 0001-8708. Available under: doi: 10.1016/j.aim.2007.09.016
Zusammenfassung

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.

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Fachgebiet (DDC)
510 Mathematik
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sum of squares, Connes’ embedding conjecture, quadratic module, tracial state, von Neumann algebra.
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ISO 690KLEP, Igor, Markus SCHWEIGHOFER, 2008. Connes' embedding conjectures and sums of hermitian squares. In: Advances in Mathematics. 2008, 217(4), pp. 1816-1837. ISSN 0001-8708. Available under: doi: 10.1016/j.aim.2007.09.016
BibTex
@article{Klep2008Conne-15621,
  year={2008},
  doi={10.1016/j.aim.2007.09.016},
  title={Connes' embedding conjectures and sums of hermitian squares},
  number={4},
  volume={217},
  issn={0001-8708},
  journal={Advances in Mathematics},
  pages={1816--1837},
  author={Klep, Igor and Schweighofer, Markus}
}
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