  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 $p=1$.
