Connes' embedding conjectures and sums of hermitian squares

Klep, Igor; Schweighofer, Markus

Connes' embedding conjectures and sums of hermitian squares

Files

connes.pdf(1.63 MB)

Date

2008

Publication type

Journal article

Published in

Advances in Mathematics ; 217 (2008), 4. - pp. 1816-1837. - ISSN 0001-8708

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 II₁-factors are considered instead of matrices. Under the presence of Connes' conjecture, we derive degree bounds for the certificates.

Subject (DDC)

510 Mathematics

Keywords

sum of squares,Connes’ embedding conjecture,quadratic module,tracial state,von Neumann algebra.

Cite This

ISO 690

KLEP, Igor, Markus SCHWEIGHOFER, 2008. Connes' embedding conjectures and sums of hermitian squares. In: Advances in Mathematics. 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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Yes