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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Quadratic modules, $ C^*$-algebras, and free convexity


Authors: Vadim Alekseev, Tim Netzer and Andreas Thom
Journal: Trans. Amer. Math. Soc. 372 (2019), 7525-7539
MSC (2010): Primary 14P10, 46L89
DOI: https://doi.org/10.1090/tran/7230
Published electronically: September 6, 2019
MathSciNet review: 4029672
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Abstract: Given a quadratic module, we construct its universal $ C^*$-algebra, and then use methods and notions from the theory of $ C^*$-algebras to study the quadratic module. We define residually finite-dimensional quadratic modules, and characterize them in various ways, in particular via a Positivstellensatz. We give unified proofs for several existing strong Positivstellensätze, and prove some new ones. Our approach also leads naturally to interesting new examples in free convexity. We show that the usual notion of a free convex hull is not able to detect residual finite-dimensionality. We thus study a notion of free convexity which is coordinate-free. We characterize semialgebraicity of free convex hulls of semialgebraic sets, and show that they are not always semialgebraic, even at scalar level. This also shows that the membership problem for quadratic modules (a well-studied problem in Real Algebraic Geometry) has a negative answer in the non-commutative setup.


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Additional Information

Vadim Alekseev
Affiliation: Institut für Geometrie, TU Dresden, 01062 Dresden, Germany
Email: vadim.alekseev@tu-dresden.de

Tim Netzer
Affiliation: Department of Mathematics, University of Innsbruck, Innsbruck, Austria
Email: tim.netzer@uibk.ac.at

Andreas Thom
Affiliation: Institut für Geometrie, TU Dresden, 01062 Dresden, Germany
Email: andreas.thom@tu-dresden.de

DOI: https://doi.org/10.1090/tran/7230
Received by editor(s): April 11, 2016
Received by editor(s) in revised form: March 7, 2017
Published electronically: September 6, 2019
Additional Notes: This research was supported by ERC Starting Grant No. 277728 and the ERC Consolidator Grant No. 681207.
Article copyright: © Copyright 2019 American Mathematical Society