. O A ] 2 0 Fe b 20 07 PROPERTY A , PARTIAL TRANSLATION STRUCTURES AND UNIFORM EMBEDDINGS IN GROUPS

We define the concept of a partial translation structure T on a metric space X and we show that there is a natural C *-algebra C * (T) associated with it which is a subalgebra of the uniform Roe algebra C * u (X). We introduce a coarse invariant of the metric which provides an obstruction to embedding the space in a group. When the space is sufficiently group-like, as determined by our invariant, properties of the Roe algebra can be deduced from those of C * (T). We also give a proof of the fact that the uniform Roe algebra of a metric space is a coarse invariant up to Morita equivalence. Many interesting geometric properties of spaces and groups are captured by the structure of C *-algebras associated with those objects. For example, a discrete group G is amenable if and only if the full C *-algebra C * (G) is nuclear [7]. In a similar vein, for a discrete group G, Yu's property A is equivalent both to the nuclearity of the uniform Roe algebra C * u (G) and to the exactness of the reduced C *-algebra C While property A and the uniform Roe algebra can be defined for arbitrary metric spaces, we cannot generalise these results without a good analogue of the reduced C *-algebra of a group. In this paper we introduce a C *-algebra to fulfill this role. To do so we carry out the following programme. First we define the notion of a partial translation structure (Definition 11) on a uniformly discrete metric space, which captures geometrically the interplay between the left and the right action of a group on itself. In broad terms, this can be described as follows. In Euclidean space translations