At the very end of my book Lectures on Coarse Geometry I asked the following question: suppose you take a discrete group \( \Gamma \), consider it as a metric space and form the uniform translation algebra \( UC^*(|\Gamma|) \). This algebra has a natural \( \Gamma \)-action and the \( \Gamma \)-fixed subalgebra, \( UC^*(|\Gamma|)^\Gamma \), clearly contains the reduced \( C^* \)-algebra of the group \( \Gamma \). Are these objects equal? In the book I showed that they are equal for amenable groups and outlined an argument, invented by Nigel Higson, which shows that they are also equal for free groups – this uses Haagerup’s results about rapid decay.
It is clear that some kind of approximation property is involved here and in the book I called it the “invariant translation approximation property”. (In our earlier discussions Nigel, Jerry and I were so irritated by this question that we called it the “completely stupid approximation property” but fortunately we were not completely stupid enough to use this term in print. Ooops…) Which groups possess this property?