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 $SL(3,Z)$ 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.

## 2 comments:

Just wanted to add that recently, Lafforgue and de la Salle have actually shown that any lattice subgroup $\Gamma$ of $SL(n,F)$, where $n > 2$ and $F$ is a local non-archimedian field, has not the OAP (math.OA.1003.2327). Since such $\Gamma$ is certainly exact this provides examples of groups without the strengthened ITAP. In particular there is thus a closed subspace $S$ of the compact operators on $l_2$ such that the set of invariant elements of $UC^*(\Gamma;S)$ is not equal to $C^*_r(\Gamma)\otimes S$.

Thank you very much, Joachim!

Just for clarity, the reference to Lafforgue and de la Salle is 1004.2327 (not 1003.2327).

Vincent Lafforgue and Mikael de la Salle, “Non commutative Lp spaces without the completely bounded approximation property,” 1004.2327 (April 14, 2010), http://arxiv.org/abs/1004.2327.

Post a Comment