2016/02/04 by V. G. Alekseev, Tim Netzer, Alekseev, Vadim +3
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.1602.01618
openalex publication_date 2016/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 propose a new 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 has a negative answer in the non-commutative setup.