Inductive limits of partial crossed products
Md Amir Hossain
TL;DR
The paper develops a framework for inductive limits of partial dynamical systems, proving that the inductive limit partial action on $A=\varinjlim A^{(i)}$ yields a partial crossed product $A\rtimes_α G$ canonically isomorphic to the inductive limit of the individual crossed products. It then presents three applications: globalization of the inductive limit action when each stage is globalizable, finite Rokhlin-dimension bounds for the limit (including commuting towers), and the construction of invariant tracial states on the limit crossed product via inductive limits of traces. The results provide a robust method to analyze and construct crossed products through inductive limits, with implications for structure and trace theory in noncommutative dynamics. Overall, the work connects inductive-limit techniques to partial actions, enabling computation and analysis of the limit crossed product from its finite-stage counterparts.
Abstract
Let $\big((A^{(i)}, G, α^{(i)}), φ_i\big)_{i \in \mathbb{N}}$ be an inductive sequence of partial dynamical systems. We prove the existence of an induced partial action $α$ of $G$ on the inductive limit $A=\varinjlim A^{(i)}$. We call $α$ the inductive limit partial action. Furthermore, we show the corresponding partial crossed product $A\rtimes_αG$ is canonically isomorphic to $\varinjlim A^{(i)}\rtimes_{α^{(i)}}G$. We also study the globalization of the inductive limit partial action $α$, its finite Rokhlin dimension and tracial states on $A\rtimes_αG$.