Model Theory of General von Neumann Algebras II: Group Actions and Crossed Products

TL;DR

This paper develops a model-theoretic framework for W*-dynamical systems, focusing on continuous group actions on von Neumann algebras, and explores their ultraproducts, crossed products, and computability aspects.

Contribution

It axiomatizes continuous group actions on von Neumann algebras and demonstrates how ultraproducts and crossed products interact within this framework, linking to computable dynamics.

Findings

Ultraproducts of continuous actions commute with crossed products.

Crossed product construction aids in producing computable presentations.

Connections established between model theory, computability, and von Neumann algebra dynamics.

Abstract

Expanding on previous work of the author, we initiate the model theoretic study of W$^*$-dynamical systems. We axiomatize continuous weight-preserving group actions of $G$ on von Neumann algebras for $G$ a given locally compact Hausdorff group. Since our axiomatization is of continuous actions, the ultraproduct is defined so that the ultraproduct action of $G$ is also continuous. Building on a theorem of Tomatsu, we show that continuous ultraproducts commute with crossed products. Finally, we prove a suite of results about computability of the aforementioned axiomatizations and of presentations of crossed products. In particular, we show how the crossed product construction is a useful tool for producing computable presentations, giving special attention to the group measure space construction of Murray and von Neumann. Thereby, we establish interesting connections to computable dynamics and computable measure theory.