# Examples of groups without the n-positive approximation property

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 $n + 1$?

• Very interesting question, (Although I was a bit surprised to see $M(G)$ denoting the group von Neumann algebra, not that this causes any real confusion here.) Jul 16 '10 at 17:29
• Yes, I think perhaps $W^*(G)$ or $L(G)$ is more common. I took the notation from Cowling and Haagerup's paper. Jul 16 '10 at 18:04
• Jesse, if you mean the 1989 Inventiones paper, then it looks to me like they denote the group von Neumann algebra by VN(G) and use M(G) for the multipliers of A(G)? Jul 16 '10 at 19:38
• Ah, I see that de Canniere and Haagerup use ${\mathcal M}(G)$ for the group von Neumann algebra. I guess one is just doomed to conflicts of notation, whatever one chooses ... :-) Jul 16 '10 at 20:00
• Yes, that's the one I mean, thank you. Jul 16 '10 at 20:04

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.