Let $G$ be a locally compact group and let $A(G)$ be the http://eom.springer.de/f/f120080.htm>Fourier algebra of $G$, which we view as the predual of the group von Neumann algebra $\mathcal M(G)$. Let $MA(G)$ be the space of multipliers of $A(G)$, i.e., $\varphi \in MA(G)$ if and only if $\varphi \psi \in A(G)$ for all $\psi \in A(G)$. Then $\varphi \in MA(G)$ induces a bounded operator $m\_\varphi: A(G) \rightarrow A(G)$, and hence also a bounded operator $M\_\varphi = m\_\varphi^\*$ on $\mathcal M(G)$. 


$M\_\varphi$ is completely bounded if $\| M\_\varphi \|\_{CB} = \sup\_n \| M\_\varphi \otimes {\rm id}\_n \| < \infty$, where ${\rm id}\_n$ is the identity operator on the $n \times n$ matrices $\mathbb M\_n(\mathbb C)$. $M\_\varphi$ is $n$-positive if $M\_\varphi \otimes {\rm id}\_n$ takes the positive cone $\mathcal M(G)\_+$ into itself, or equivalently $\| M\_\varphi \otimes {\rm id}\_n \| = \varphi(e)$. $M\_\varphi$ is completely positive if it is $n$-positive for every $n \in \mathcal N$.


A well known result is that $G$ is amenable if and only if $A(G)$ has an approximate unit $\{ \varphi\_k \}\_k$ such that $M\_{\varphi\_k}$ is completely positive for all $k$. Haagerup showed that $SL\_2(\mathbb R)$, and all of its lattices have the completely bounded approximation property: For these groups, $A(G)$ has an approximate unit $\{ \varphi\_k \}\_k$ such that $\sup\_k \| M\_{\varphi\_k} \|\_{CB} < \infty$, (in fact he showed that this supremum can be 1 for $SL\_2(\mathbb R)$, and all of its lattices). To contrast, he also showed that $SL\_m(\mathbb R)$, and all of its lattices do not have the completely bounded approximation property whenever $m \geq 3$. De Canniere and Haagerup have also shown that free groups have the $n$-positive approximation property for every $n \in \mathbb N$: For these groups, given any $n \in \mathbb N$, $A(G)$ has an approximate unit $\{ \varphi\_k \}$ of compactly supported functions such that $\varphi\_k$ is $n$-positive.


Recently, I was at a conference and Mikael de la Salle asked me if I knew of any examples of groups which do not have the $n$-positive approximation property. I do not and so I thought I would ask here.


1) What is an example of a group which for some $n$ does not have the $n$-positive approximation property?


2) What is an example of a group which for any $n$ does not have the $n$-positive approximation property?


3) Are there groups which have the $n$-positive approximation property for some $n$, but not for $n + 1$?



Examples of groups without the n-positive approximation property
Haagerup actually proved (lattices of) higher rank Lie groups do not have $1$-positive approximation property, nor bounded approximation property, i.e., there is no uniformly bounded sequence of compactly supported multipliers that converges pointwise to $1$. (It's still open whether the reduced group C$^\*$-algebra of such a group also fails bounded approximation property.) Also, lamplighter groups on non-amenable groups do not have $1$-positive approximation property (bounded approximation property with constant $1$). The reason is that the proofs of no $1$-positive approximation property boil down to non-existence of certain types of multipliers on an amenable group; and for multipliers $M\_{\varphi}$ on an amenable group, cb-norm coincides with norm. (See Cowling et al., Duke Math. J. 127 (2005), 429--486 for a survey.) Problem 3 seems inaccessible. 


