A quasi-local characterisation of $L^p$-Roe algebras

TL;DR

This paper extends the quasi-local characterisation of Roe algebras to the $L^p$ setting for all $p$ in [1,∞), addressing challenges especially for the $p=1$ case, thereby broadening the applicability of the theory.

Contribution

It generalizes the quasi-local characterisation of (uniform) Roe algebras to $L^p$-spaces for all $p$, including the non-reflexive case of $p=1$, improving previous methods.

Findings

Extended quasi-local characterisation to $L^p$-Roe algebras for all $p$

Addressed technical challenges for $p=1$ due to lack of reflexivity

Broadened the applicability of Roe algebra characterisations

Abstract

Very recently, \v{S}pakula and Tikuisis provide a new characterisation of (uniform) Roe algebras via quasi-locality when the underlying metric spaces have straight finite decomposition complexity. In this paper, we improve their method to deal with the $L^p$-version of (uniform) Roe algebras for any $p\in [1,\infty)$. Due to the lack of reflexivity on $L^1$-spaces, some extra work is required for the case of $p=1$.