M ar 2 00 6 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 * r (G). This follows from the results of Anantharaman-Delaroche and Renault [1], Higson and Roe [5], Guentner and Kaminker [4], and Ozawa [9]. 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 are distinguished from other isometries of the space by the fact that they move each point by the same distance. Let us assume that a group G is equipped with a left invariant metric d, which means that for any elements g, s, t of G, d(gs, gt) = d(s, t). In other words, multiplication on the left acts by isometries on the metric space (G, d). On the other hand, the right multiplication by a fixed element g of G moves each …