In his work on the Novikov conjecture Yu introduced an non- generalisation of amenability, Yu's Property A, in which equivariance is replaced by a controlled support condition which captures more of the geometry. Spaces satisfying Yu's condition also satisfy the Coarse Baum Connes conjecture.
There are several well known homological characterisations of amenability and Higson asked if there are analogous characterisations of property A. We have answered this question in several different ways. In the context of discrete metric spaces I, together with Wright and Brodzki, introduced an analogue of bounded cohomology which characterises Yu’s property A in terms of vanishing, and provides the notion of an asymptotically invariant mean for a space. In the context of topological actions we introduced, with Nowak, a class of Banach modules associated to the action such that the vanishing of group cohomology over these coefficients characterises topological amenability of the action. Specialising to the action of a group on its Stone Cech compactification we obtain a characterisaction of C* exactness for the group generalising Johnson’s celebrated characterisation of amenability in terms of vanishing bounded cohomology. An alternative homological view of amenability was provided by Block and Weinberger who showed that for a metric space amenability could be characterised by non vanishing uniformly finite homology. In recent work we dualised the cohomology theory described above to provide a notion of the uf homology of a topological action providing a characterisation of topological amenability in terms of the non vanishing of the first uf homology. This dualisation arose from an earlier study by Brodzki, Niblo and Wright of the relationship between classical bounded cohomology and uniformly finite homology for a group.