Trace-positive polynomials and the quartic tracial moment problem |
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Authors: | Sabine Burgdorf Igor Klep |
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Institution: | 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, Koro?ka 160, 2000 Maribor, Slovenia;4. Univerza v Ljubljani, Fakulteta za matematiko in fiziko, Jadranska 19, 1000 Ljubljana, Slovenia |
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Abstract: | 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. |
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