Quadratic Modules, C*-Algebras, and Free Convexity

TL;DR

This paper links quadratic modules with C*-algebras to explore free convexity, introducing new concepts and results, including a coordinate-free free convexity notion and insights into semialgebraic sets.

Contribution

It constructs universal C*-algebras from quadratic modules, characterizes residually finite-dimensional modules, and proposes a new coordinate-free free convexity framework.

Findings

Unified proofs for strong Positivstellens"atze

New examples in free convexity

Free convex hulls may not be semialgebraic

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 Positivstellens\"atze, 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.