  In this paper, we study the relation between the uniform Roe algebra and the
uniform quasi-local algebra associated to a metric space of bounded geometry.
In the process, we introduce a weakening of the notion of expanders, called
asymptotic expanders. We show that being a sequence of asymptotic expanders is
a coarse property under certain connectedness condition, and it implies
non-uniformly local amenability. Moreover, we also analyse some $C^*$-algebraic
properties of uniform quasi-local algebras. In particular, we show that a
uniform quasi-local algebra is nuclear if and only if the underlying metric
space has Property A.
