Soft inductive limits of operator systems and a noncommutative Lazar-Lindenstrauss theorem
Kristin Courtney, Niklas Galke, Lauritz van Luijk, Alexander Stottmeister
TL;DR
The paper generalizes inductive limits to soft inductive systems in the operator-system framework by relaxing coherence to an asymptotic condition and indexing over nets. It shows that every nuclear operator system is a soft inductive limit of nets of finite-dimensional von Neumann algebras, and in the separable case this is equivalent to being a strict inductive limit; this yields a noncommutative Lazar-Lindenstrauss theorem via noncommutative Choquet duality. The authors connect completely positive approximation properties with soft inductive limits, leveraging Ding–Peterson arguments to obtain the strict-limit characterization. They further translate these results into noncommutative Choquet theory, identifying nuclear nc simplices as projective limits of nc state spaces of finite-dimensional von Neumann algebras. The framework provides a structural description of nuclear operator systems and their nc state spaces, with potential implications for quantum theory and noncommutative convexity.
Abstract
We establish a flexible generalization of inductive systems of operator systems, which relaxes the usual transitivity (or coherence) condition to an asymptotic version thereof and allows for systems indexed over arbitrary nets. To illustrate the utility of this generalization, we highlight how such systems arise naturally from completely positive approximations of nuclear operator systems. Going further, we utilize an argument of Ding and Peterson to show that a separable operator system is nuclear if and only if it is an inductive limit of matrix algebras, generalizing a classic Theorem of Lazar and Lindenstrauss to the setting of noncommutative Choquet theory.