Quasi-local algebras and asymptotic expanders

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