Invariant translation approximation

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?

While talking with Nowak in Texas I learned about a paper by Joachim Zacharias, On the invariant translation approximation property for discrete groups , which makes significant progress on this question. Zacharias’ paper works as follows. First consider a strengthening of the ITAP by allowing coefficients: one looks at \( UC^*(|\Gamma|;S) \) where \( S \) is an auxiliary \( C^* \)-algebra (or operator space) and asks whether the \( \Gamma \)-invariant part of that is equal to \( C^*_r(\Gamma)\otimes S \) (minimal tensor product). (N.B. There is no \( \Gamma \)-action on \( S \) – no ‘twisting’.) Zacharias proves that for exact groups this strengthened ITAP is equivalent to the Haagerup-Kraus approximation property ( Approximation properties for group \( C^* \)-algebras and group von Neumann algebras , Transactions of the American Mathematical Society, Vol. 344, No. 2 (Aug., 1994), pp. 667-699, which says that there is a net in the Fourier algebra \( A(\Gamma) \) converging to 1 in a certain weak topology on the completely bounded multipliers on \( C^*_r(\Gamma) \). Unfortunately no example of an exact discrete group without this property is known, but it has been conjectured that \( SL(3,{\mathbb Z}) \) is such a group.

On the way the author proves another characterization of exact groups, namely that \( \Gamma \) is exact iff the map \( S \mapsto UC^*(|\Gamma|;S) \) is an exact functor.

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