# 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 $n + 1$?

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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.) –  Yemon Choi 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. –  Jesse Peterson 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)? –  Yemon Choi 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 ... :-) –  Yemon Choi Jul 16 '10 at 20:00
Yes, that's the one I mean, thank you. –  Jesse Peterson 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.