Inductive limits of partial crossed products

TL;DR

This paper constructs an inductive limit of partial dynamical systems and shows the associated crossed product is isomorphic to the inductive limit of the individual crossed products, also exploring globalization and Rokhlin properties.

Contribution

It introduces the concept of inductive limit partial actions and proves the isomorphism of the crossed product with the inductive limit of crossed products, extending the theory of partial dynamical systems.

Findings

Existence of an induced partial action on the inductive limit.

Isomorphism between the crossed product of the inductive limit and the limit of crossed products.

Analysis of globalization, Rokhlin dimension, and tracial states on the crossed product.

Abstract

Let $\big((A^{(i)}, G, \alpha^{(i)}), \phi_i\big)_{i \in \mathbb{N}}$ be an inductive sequence of partial dynamical systems. We prove the existence of an induced partial action $\alpha$ of $G$ on the inductive limit $A=\varinjlim A^{(i)}$. We call $\alpha$ the inductive limit partial action. Furthermore, we show the corresponding partial crossed product $A\rtimes_{\alpha}G$ is canonically isomorphic to $\varinjlim A^{(i)}\rtimes_{\alpha^{(i)}}G$. We also study the globalization of the inductive limit partial action $\alpha$, its finite Rokhlin dimension and tracial states on $A\rtimes_{\alpha}G$.