Quadratic Modules, C*-Algebras, and Free Convexity

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 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.

References in corpus (2)

• Free convex sets defined by rational expressions have LMI representations
• Free Convex Algebraic Geometry

Cited by in corpus (4)

• Geometry of free loci and factorization of noncommutative polynomials
• Dilation theory in finite dimensions and matrix convexity
• Globally trace-positive noncommutative polynomials and the unbounded tracial moment problem
• Quantum Information Theory and Free Semialgebraic Geometry: One Wonderland Through Two Looking Glasses