Globally trace-positive noncommutative polynomials and the unbounded tracial moment problem

A noncommutative (nc ) polynomial is called (globally) trace-positive 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 physi...

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Vydáno v:Mathematische annalen Ročník 387; číslo 3-4; s. 1403 - 1433
Hlavní autoři: Klep, Igor, Scheiderer, Claus, Volčič, Jurij
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2023
Springer Nature B.V
Témata:
Algebra
Bivariate analysis
Commutators
Hilbert space
Mathematical analysis
Mathematics
Mathematics and Statistics
Operators (mathematics)
Optimization
Polynomials
Rational functions
Sums
46L51
Primary 13J30
47L60
14P99 Secondary 47A63
ISSN:0025-5831, 1432-1807
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Shrnutí:A noncommutative (nc ) polynomial is called (globally) trace-positive 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, trace-positive 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 sum-of-squares certificate with concrete univariate denominators for nonnegative bivariate polynomials. The trace positivity certificates in this paper are obtained by convex duality through solving the so-called 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 trace-positive nc polynomial admits an explicit approximation in the 1-norm on its coefficients by sums of hermitian squares and commutators of nc polynomials.
Bibliografie:ObjectType-Article-1
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ISSN:0025-5831
1432-1807
DOI:10.1007/s00208-022-02495-5