(Note to those who like to tidy LaTeX, or ${\rm \LaTeX}$: I kindly request that you don't put any LaTeX in the title of this question, nor change the bolds below to blackboard bold.)$\newcommand{\FA}{{\rm A}}$

The question in the title might be an open problem, but the people I've consulted aren't sure. In any case there are several people here who might have thought about related problems that would shed light on this, and raising the question on MO seems more efficient and more sociable than emailing these people en masse.

To be more precise: does there exist a net $(u_\alpha)$ in $A:=\FA({\rm SL}_3({\bf Z}))$, not necessarily bounded, such that $$\Vert au_\alpha -a \Vert_A \to 0 \quad\hbox{for each $a\in A$?} $$

Feel free to replace ${\rm SL}_3({\bf Z})$ by other lattices in higher rank Lie groups, if you think that helps. It's known that if such a net exists then it can't be bounded in the cb-multiplier norm

Context: in the paper of Haagerup and Kraus which introduces the (H)AP for groups (Trans. AMS 1994), it follows from Theorem 1.11 that if a locally compact group $G$ has the (H)AP, its Fourier algebra has an approximate identity. There is a somewhat technical partial converse, but it's not clear to me if the result is expected to be if and only if. Anyway, it has recently been shown through work of Lafforgue and de la Salle that SL(3,Z) and some other lattices in higher rank Lie groups do not have the (H)AP, hence this question.

  • $\begingroup$ What do you mean by "somewhat technical partial converse"? Just Thm 1.11, namely some sort of tensoring against a group with a sufficient rich set of representations? $\endgroup$ – Matthew Daws Mar 5 '14 at 9:58
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    $\begingroup$ To my knowledge, the problem is open. The approach of Lafforgue and de la Salle to prove that $\mathrm{SL}(3,\mathbb{R})$ (or $\mathrm{SL}(3,\mathbb{Z})$) does not have the AP, and the methods of Haagerup and myself to extend this result to (lattices in) higher rank simple Lie groups use relatively strong asymptotic behaviour for, in the case of $\mathrm{SL}(3,\mathbb{R})$, the $\mathrm{SO}(3)$-bi-invariant elements of the Fourier algebra of $\mathrm{SL}(3,\mathbb{R})$. Perhaps this could be helpful when considering possible stronger results than non-AP. $\endgroup$ – Tim de Laat Mar 5 '14 at 10:41
  • $\begingroup$ @MatthewDaws Yes, that's exactly what I meant but I was too lazy to copy out the precise details at 3am $\endgroup$ – Yemon Choi Mar 5 '14 at 14:15

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