Publikation: Trace-positive polynomials and the quartic tracial moment problem
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Zusammenfassung
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.
Zusammenfassung in einer weiteren Sprache
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.
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BURGDORF, Sabine, Igor KLEP, 2010. Trace-positive polynomials and the quartic tracial moment problem. In: Comptes Rendus Mathematique. Elsevier. 2010, 348(13-14), pp. 721-726. ISSN 1631-073X. eISSN 1778-3569. Available under: doi: 10.1016/j.crma.2010.06.005BibTex
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year={2010},
doi={10.1016/j.crma.2010.06.005},
title={Trace-positive polynomials and the quartic tracial moment problem},
number={13-14},
volume={348},
issn={1631-073X},
journal={Comptes Rendus Mathematique},
pages={721--726},
author={Burgdorf, Sabine and Klep, Igor}
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