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Connes' embedding conjectures and sums of hermitian squares

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2008

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http://nbn-resolving.de/urn:nbn:de:bsz:352-156210
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https://doi.org/10.1016/j.aim.2007.09.016
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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.

Zusammenfassung in einer weiteren Sprache

Fachgebiet (DDC)
510 Mathematik

Schlagwörter

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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