Trace-positive polynomials and the quartic tracial moment problem

Sabine Burgdorf; Igor Klep

doi:10.1016/j.crma.2010.06.005

Algebra/Functional Analysis

Trace-positive polynomials and the quartic tracial moment problem
[Polynômes avec une trace positive et le problème quartique des moments traciaux]

Présenté par : Gilles Pisier

Sabine Burgdorf ^1,² ; Igor Klep ^3,⁴

¹ Institut de recherche mathématique de Rennes, université de Rennes 1, campus de Beaulieu, 35042 Rennes cedex, France
² Universität Konstanz, Fachbereich Mathematik und Statistik, 78457 Konstanz, Germany
³ Univerza v Mariboru, Fakulteta za naravoslovje in matematiko, Koroška 160, 2000 Maribor, Slovenia
⁴ Univerza v Ljubljani, Fakulteta za matematiko in fiziko, Jadranska 19, 1000 Ljubljana, Slovenia

Comptes Rendus. Mathématique, Volume 348 (2010) no. 13-14, pp. 721-726.

Résumé
Abstract

Nous présentons l'analogue tracial du résultat classique de Hilbert sur les quartiques positives : un polynôme de degré quatre en deux variables non commutatives ayant une trace positive est une somme de carrés hermitiens et de commutateurs. Ceci est appliqué par dualité à l'étude du problème tronqué des moments traciaux : une suite de nombres réels indexée par des mots de degré quatre en deux variables non commutatives, ayant des valeurs invariantes par permutations circulaires des indices, peut être représentée par des moments traciaux, si la matrice des moments est définie positive.

The tracial analog of Hilbert's classical result on positive binary quartics is presented: a trace-positive bivariate noncommutative polynomial of degree at most four is a sum of hermitian squares and commutators. This is applied via duality to investigate the truncated tracial moment problem: a sequence of real numbers indexed by words of degree four in two noncommuting variables with values invariant under cyclic permutations of the indexes, can be represented with tracial moments of matrices if the corresponding moment matrix is positive definite. Understanding trace-positive polynomials and the tracial moment problem is one of the approaches to Connes' embedding conjecture.

Reçu le : 2010-03-24
Accepté le : 2010-06-02
Publié le : 2010-06-23

DOI : 10.1016/j.crma.2010.06.005

Affiliations des auteurs :

Sabine Burgdorf ^{1, 2} ; Igor Klep ^{3, 4}

¹ Institut de recherche mathématique de Rennes, université de Rennes 1, campus de Beaulieu, 35042 Rennes cedex, France
² Universität Konstanz, Fachbereich Mathematik und Statistik, 78457 Konstanz, Germany
³ Univerza v Mariboru, Fakulteta za naravoslovje in matematiko, Koroška 160, 2000 Maribor, Slovenia
⁴ Univerza v Ljubljani, Fakulteta za matematiko in fiziko, Jadranska 19, 1000 Ljubljana, Slovenia

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     title = {Trace-positive polynomials and the quartic tracial moment problem},
     journal = {Comptes Rendus. Math\'ematique},
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     year = {2010},
     doi = {10.1016/j.crma.2010.06.005},
     language = {en},
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Sabine Burgdorf; Igor Klep. Trace-positive polynomials and the quartic tracial moment problem. Comptes Rendus. Mathématique, Volume 348 (2010) no. 13-14, pp. 721-726. doi : 10.1016/j.crma.2010.06.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.06.005/

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