Quasi-Locality and Property A

TL;DR

This paper proves that quasi-local operators on certain Banach space-valued sequence spaces belong to Roe algebras under specific geometric and functional conditions, extending previous results and characterizing Property A.

Contribution

It generalizes existing results by establishing inclusion of quasi-local operators in Roe algebras for spaces with Property A and various p-values, and provides new characterizations of Property A.

Findings

Quasi-local operators are in Roe algebras under Property A for p in (1,∞).

Uniform ℓ^p-Roe algebras are inverse-closed for spaces with Property A.

A new operator norm localization condition characterizes Property A.

Abstract

Let $X$ be a metric space with bounded geometry, $p\in\{0\} \cup [1,\infty]$, and let $E$ be a Banach space. The main result of this paper is that either if $X$ has Yu's Property A and $p\in(1,\infty)$, or without any condition on $X$ when $p\in\{0,1,\infty\}$, then quasi-local operators on $\ell^p(X,E)$ belong to (the appropriate variant of) 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 $\ell^p$-Roe algebras (of spaces with Property A) are closed under taking inverses, and another condition characterising Property A, akin to operator norm localisation for quasi-local operators.