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 -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.
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Kang Li, Piotr W. Nowak, Ján Špakula, Jiawen Zhang, Quasi-local algebras and asymptotic expanders. Groups Geom. Dyn. 15 (2021), no. 2, pp. 655–682DOI 10.4171/GGD/610