K-theory for the maximal Roe algebra of certain expanders

We study in this paper the maximal version of the coarse Baum–Connes assembly map for families of expanding graphs arising from residually finite groups. Unlike for the usual Roe algebra, we show that this assembly map is closely related to the (maximal) Baum–Connes assembly map for the group and is an isomorphism for a class of expanders. We also introduce a quantitative Baum–Connes assembly map and discuss its connections to K-theory of (maximal) Roe algebras.