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

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

No Thumbnail Available

##### Files

There are no files associated with this item.

##### Date

2022

##### Authors

##### Editors

##### Journal ISSN

##### Electronic ISSN

##### ISBN

##### Bibliographical data

##### Publisher

##### Series

##### DOI (citable link)

##### International patent number

##### Link to the license

oops

##### EU project number

##### Project

##### Open Access publication

##### Collections

##### Title in another language

##### Publication type

Journal article

##### Publication status

Published

##### Published in

Mathematische Annalen ; 2022. - Springer. - ISSN 0025-5831. - eISSN 1432-1807

##### Abstract

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.

##### Summary in another language

##### Subject (DDC)

510 Mathematics

##### Keywords

##### Conference

##### Review

undefined / . - undefined, undefined. - (undefined; undefined)

##### Cite This

## ISO 690

KLEP, Igor, Claus SCHEIDERER, Jurij VOLČIČ, 2022.*Globally trace-positive noncommutative polynomials and the unbounded tracial moment problem*. In: Mathematische Annalen. Springer. ISSN 0025-5831. eISSN 1432-1807. Available under: doi: 10.1007/s00208-022-02495-5

## BibTex

@article{Klep2022-10Globa-58993, year={2022}, doi={10.1007/s00208-022-02495-5}, title={Globally trace-positive noncommutative polynomials and the unbounded tracial moment problem}, issn={0025-5831}, journal={Mathematische Annalen}, author={Klep, Igor and Scheiderer, Claus and Volčič, Jurij} }

## RDF

<rdf:RDF xmlns:dcterms="http://purl.org/dc/terms/" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:bibo="http://purl.org/ontology/bibo/" xmlns:dspace="http://digital-repositories.org/ontologies/dspace/0.1.0#" xmlns:foaf="http://xmlns.com/foaf/0.1/" xmlns:void="http://rdfs.org/ns/void#" xmlns:xsd="http://www.w3.org/2001/XMLSchema#" > <rdf:Description rdf:about="https://kops.uni-konstanz.de/server/rdf/resource/123456789/58993"> <dcterms:title>Globally trace-positive noncommutative polynomials and the unbounded tracial moment problem</dcterms:title> <dc:creator>Scheiderer, Claus</dc:creator> <dc:contributor>Klep, Igor</dc:contributor> <foaf:homepage rdf:resource="http://localhost:8080/"/> <dcterms:abstract xml:lang="eng">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.</dcterms:abstract> <bibo:uri rdf:resource="https://kops.uni-konstanz.de/handle/123456789/58993"/> <dc:contributor>Scheiderer, Claus</dc:contributor> <dc:creator>Volčič, Jurij</dc:creator> <dcterms:issued>2022-10</dcterms:issued> <dc:date rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2022-11-02T08:11:44Z</dc:date> <dc:creator>Klep, Igor</dc:creator> <dc:language>eng</dc:language> <void:sparqlEndpoint rdf:resource="http://localhost/fuseki/dspace/sparql"/> <dspace:isPartOfCollection rdf:resource="https://kops.uni-konstanz.de/server/rdf/resource/123456789/39"/> <dcterms:isPartOf rdf:resource="https://kops.uni-konstanz.de/server/rdf/resource/123456789/39"/> <dcterms:available rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2022-11-02T08:11:44Z</dcterms:available> <dc:contributor>Volčič, Jurij</dc:contributor> </rdf:Description> </rdf:RDF>

##### Internal note

##### xmlui.Submission.submit.DescribeStep.inputForms.label.kops_note_fromSubmitter

##### Examination date of dissertation

##### Method of financing

##### Comment on publication

##### Alliance license

##### Corresponding Authors der Uni Konstanz vorhanden

##### International Co-Authors

##### Bibliography of Konstanz

Yes

##### Refereed

Yes

Online First: Journal articles that are published online before they appear as an actual part of a journal issue. Online first articles are published on the journal's website in the publisher's version.