Connes' embedding conjecture and sums of hermitian squares

By Sabine Burgdorf, K. Dykema, Igor Klep, Markus Schweighofer. 2006

100 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/83332e0869fe9080a22b0401855f7f9363085b1f

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In Algebra and Number Theory

1. Benoît Collins
2. Tobias Fritz
3. Édouard Maurel-Segala
4. J. William Helton
5. Mihai Putinar
6. Nathanial P. Brown Narutaka Ozawa
7. Alice Guionnet
8. Gilles Pisier

In Geometry and Topology

1. Alice Guionnet
2. Édouard Maurel-Segala
3. J. William Helton
4. Mihai Putinar
5. Nathanial P. Brown Narutaka Ozawa
6. Benoît Collins
7. Roland Speicher
8. E. Guentner

In Mathematical Physics

1. Édouard Maurel-Segala
2. Benoît Collins
3. Nathanial P. Brown Narutaka Ozawa
4. Alice Guionnet
5. Tobias Fritz
6. J. William Helton
7. Mihai Putinar
8. Gilles Pisier

Citations in the corpus

• Asymptotics of unitary and orthogonal matrix integrals (2006, background): “i )1≤i≤m) vanishes, which insures that V − V ∗ = P l PlQl − QlPl for some polynomials Pl,Ql, cf [9] Lemma 2.9 6 BENOˆIT COLLINS, ALICE GUIONNET, AND EDOUARD MAUREL-SEGALA for a probabilistic proof or [22], Proposition 2.3 for a direct proof (in the real symmetric case, but directly adaptable to the ”
• TSIRELSON'S PROBLEM AND KIRCHBERG'S CONJECTURE (2010, background): “s known to be equivalent to QWEP; just to mention a few, the list of equivalent open questions contains Connes’ embedding problem [Con76, Cap10], a problem on positivity of noncommutative polynomials [KS08] or on tensor products of operator systems [Kav11]. More recently, also connections to quantum”
• About the Connes Embedding Conjecture---Algebraic approaches--- (2012, methodology): “cally, we treat some easy parts of Positivstellens¨atze of Putinar ([Pu]), Helton–McCullough ([HM]), and Schmu¨dgen–Bakonyi–Timotin ([BT]). We then treat their tracial analogue by Klep– Schweighofer ([KS]), which is equivalent to the Connes Embedding Conjecture. We give new proofs of Kirchberg’s the”
• POSITIVE POLYNOMIALS IN SCALAR AND MATRIX VARIABLES, THE SPECTRAL THEOREM AND OPTIMIZATION (2006, background): “ theorem and the corollary follow the same pattern we are by now familiar with: a convex separation argument followed by a GNS construction. See for details [Radul04], and for a last minute refinement [KS06]. 74 J. WILLIAM HELTON AND MIHAI PUTINAR 9. Convexity in a free algebra Convexity of functions”