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

Jananan Arulseelan

TL;DR

The paper develops a model-theoretic framework for $G$-actions on von Neumann algebras, introducing continuous Ocneanu ultraproducts and a systematic axiomatization for both discrete and locally compact group actions preserving a faithful semifinite weight. It proves that, in the abelian case, continuous ultraproducts commute with crossed products and establishes Takesaki duality within this continuous logic setting. A central contribution is linking computability with crossed products, providing computable axiomatizations and constructions of computable presentations for crossed products and group-measure-space constructions. The results illuminate how computable dynamics and computable measure theory interplay with operator-algebraic dynamical systems, with implications for the computability of standard constructions in W*-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.

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

TL;DR

The paper develops a model-theoretic framework for -actions on von Neumann algebras, introducing continuous Ocneanu ultraproducts and a systematic axiomatization for both discrete and locally compact group actions preserving a faithful semifinite weight. It proves that, in the abelian case, continuous ultraproducts commute with crossed products and establishes Takesaki duality within this continuous logic setting. A central contribution is linking computability with crossed products, providing computable axiomatizations and constructions of computable presentations for crossed products and group-measure-space constructions. The results illuminate how computable dynamics and computable measure theory interplay with operator-algebraic dynamical systems, with implications for the computability of standard constructions in W*-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 on von Neumann algebras for a given locally compact Hausdorff group. Since our axiomatization is of continuous actions, the ultraproduct is defined so that the ultraproduct action of 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.

Paper Structure

This paper contains 12 sections, 25 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Background on Standard Forms
  3. Discrete Group Case
  4. Locally Compact Hausdorff Group Case
  5. Standard Forms of Crossed Products
  6. Standard Forms and Dual Weights
  7. Duality for Locally Compact Abelian Groups
  8. G-continuous Ocneanu Ultraproducts
  9. Computability
  10. Computable Axiomatizations
  11. Computable Presentations
  12. Group Measure Space Constructions and Computable Dynamics

Key Result

Theorem 1

Let $(\mathcal{M}_i, \Phi_i, (\alpha_g)_i)$ be a family of $G$-$\mathrm{W}^*$-dynamical systems indexed by $I$ where $G$ is a locally compact abelian group. Let $(\mathcal{M}_i \rtimes_{\alpha_i} G, \widehat{\Phi}_i, (\widehat{\alpha})_i)$ be the dual $\widehat{G}$-$\mathrm{W}^*$-dynamical system. T

Theorems & Definitions (49)

  • Theorem
  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • ...and 39 more