Quadratic modules, 𝐶*-algebras, and free convexity

Alekseev, Vadim; Netzer, Tim; Thom, Andreas

doi:10.1090/tran/7230

Quadratic modules, $C^*$-algebras, and free convexity
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by Vadim Alekseev, Tim Netzer and Andreas Thom PDF

Trans. Amer. Math. Soc. 372 (2019), 7525-7539 Request permission

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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