二次模、C*-代数与自由凸性
Quadratic Modules, C*-Algebras, and Free Convexity
AI总结：
本文通过构造二次模的 universal C*-代数，研究二次模的性质，定义并刻画残余有限维二次模，证明了多个强 Positivstellensatz 并提出新的自由凸性概念，展示了自由凸包的 semialgebraicity 问题。
摘要：
给定一个二次模，我们构造其 universal C*-代数，并利用 C*-代数理论的方法和概念来研究二次模。我们定义了残余有限维二次模，并通过 Positivstellensatz 等多种方式加以刻画。我们为几个已有的强 Positivstellensatz 提供了统一证明，并证明了一些新的结果。我们的方法自然地引导出自由凸性中的有趣新例子。我们显示通常的自由凸包概念无法检测残余有限维性。因此我们提出一个新的自由凸性概念，它是坐标自由的。我们刻画了自由凸包的 semialgebraicity，并展示它们并不总是 semialgebraic，即使在标量水平上。这也表明二次模的 membership 问题在非交换设置中有一个负的答案。
英文摘要：
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.
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