  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.
