Type of Publication:  Journal article 
Publication status:  Published 
Author:  Klep, Igor; Scheiderer, Claus; Volčič, Jurij 
Year of publication:  2022 
Published in:  Mathematische Annalen ; 33 (2022), 4.  Springer.  ISSN 00255831.  eISSN 14321807 
Pubmed ID:  36447816 
DOI (citable link):  https://dx.doi.org/10.1007/s00208022024955 
Summary: 
A noncommutative (nc ) polynomial is called (globally) tracepositive if its evaluation at any tuple of operators in a tracial von Neumann algebra has nonnegative trace. Such polynomials emerge as trace inequalities in several matrix or operator variables, and are widespread in mathematics and physics. This paper delivers the first Positivstellensatz for global trace positivity of nc polynomials. Analogously to Hilbert’s 17th problem in real algebraic geometry, tracepositive nc polynomials are shown to be weakly sums of hermitian squares and commutators of regular nc rational functions. In two variables, this result is strengthened further using a new sumofsquares certificate with concrete univariate denominators for nonnegative bivariate polynomials. The trace positivity certificates in this paper are obtained by convex duality through solving the socalled unbounded tracial moment problem, which arises from noncommutative integration theory and free probability. Given a linear functional on nc polynomials, the tracial moment problem asks whether it is a joint distribution of integral operators affiliated with a tracial von Neumann algebra. A counterpart to Haviland’s theorem on solvability of the tracial moment problem is established. Moreover, a variant of Carleman’s condition is shown to guarantee the existence of a solution to the tracial moment problem. Together with semidefinite optimization, this is then used to prove that every tracepositive nc polynomial admits an explicit approximation in the 1norm on its coefficients by sums of hermitian squares and commutators of nc polynomials.

Subject (DDC):  510 Mathematics 
Bibliography of Konstanz:  Yes 
Refereed:  Yes 
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KLEP, Igor, Claus SCHEIDERER, Jurij VOLČIČ, 2022. Globally tracepositive noncommutative polynomials and the unbounded tracial moment problem. In: Mathematische Annalen. Springer. 33(4). ISSN 00255831. eISSN 14321807. Available under: doi: 10.1007/s00208022024955
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