Persistence approximation property for $L^p$ operator algebras

TL;DR

This paper investigates the persistence approximation property in the context of $L^p$ operator algebras, introduces related assembly maps, and explores conditions impacting the $L^p$ Baum-Connes conjecture.

Contribution

It defines quantitative assembly maps for $L^p$ algebras and provides conditions under which the persistence approximation property holds, advancing the understanding of $L^p$ coarse geometry.

Findings

Established sufficient conditions for the persistence approximation property in $L^p$ crossed products and Roe algebras.

Connected the persistence approximation property to the $L^p$ (coarse) Baum-Connes conjecture.

Extended the framework of quantitative $K$-theory to $L^p$ operator algebras.

Abstract

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 $p\in [1,\infty)$. 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.