Inductive limits of partial crossed products

Abstract

Let $((A^{(i)},G,\hat{I}{\pm }^{(i)}),\ddot{I}{•}_i{)}_{i\in \mathbb{N}}$ be an inductive sequence of partial dynamical systems. We prove the existence of an induced partial action $\hat{I}\pm$ of $G$ on the inductive limit $A=\underrightarrow{\lim }{A}^{(i)}$. We call $\hat{I}\pm$ the inductive limit partial action. Furthermore, we show the corresponding partial crossed product $A{⋊}_{\hat{I}}\pm G$ is canonically isomorphic to $\underrightarrow{\lim }{A}^{(i)}{⋊}_{\hat{I}{\pm }^{(i)}}G$. We also study the globalization of the inductive limit partial action $\hat{I}\pm$, its finite Rokhlin dimension and tracial states on $A{⋊}_{\hat{I}}\pm G$.

10 pages; accepted in Semigroup Forum