Quasi-locality and Property A

Let X be a metric space with bounded geometry, p∈{0}∪[1,∞], and let E be a Banach space. The main result of this paper is that either if X has Yu's Property A and p∈(1,∞), or without any condition on X when p∈{0,1,∞}, then quasi-local operators on ℓp(X,E) belong to (the appropriate variant of) the Roe algebra of X. This generalises the existing results of this type by Lange and Rabinovich, Engel, Tikuisis and the first author, and Li, Wang and the second author. As consequences, we obtain that uniform ℓp-Roe algebras (of spaces with Property A) are closed under taking inverses, and another condition characterising Property A, akin to the operator norm localisation for quasi-local operators.