Property A for metric spaces has been introduced by Guoliang Yu as a weaker form of the Folner condition characterizing amenable groups. This property admits a number of equivalent formulations, and it can be described in terms of certain operator algebras associated with the space. For example, a discrete group G satisfies property A if and only if its reduced C*-algebra is exact.
In this talk we introduce the notion of a partial translation structure T on a metric space X, which provides an analogue of a left-right action of a group on itself. We associate a C*-algebra C*(T), which is a subalgebra of the uniform Roe algebra of X, and use it to relate the exactness of the uniform Roe algebra of X to property A. We introduce an invariant of metric spaces which provides an obstruction to the existence of a uniform embedding in a group.
This talk reports on a joint work with Graham A. Niblo and Nick Wright.