Inductive limits of partial crossed products

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 $A\rtimes_{\alpha}G$.