Persistence approximation property for $L^p$ operator algebras

In this paper, we study the persistence approximation property for quantitative $K$-theory of filtered $L^p$ operator algebras. Moreover, we define quantitative assembly maps for $L^p$ operator algebras when $p\in [1,\mathrm{\infty })$. Finally, in the case of $L^p$ crossed products and $L^p$ Roe algebras, we find sufficient conditions for the persistence approximation property. This allows us to give some applications involving the $L^p$ (coarse) Baum-Connes conjecture.