A type $II\_{1}$ factor $M$ with trace $\tau$ has Property $\Gamma$ if for every finite subset $\{ x\_{1}, x\_{2},..., x\_{n} \} \subseteq M$ and each $\epsilon >0$, there is a unitary element $u$ in $M$ with $\tau (u)=0$ and $||ux\_{j}-x\_{j}u||\_{2}<\epsilon$ for all $1 \leq j \leq n$. (Here $||T||\_2=(\tau(T^{\*}T))^{1/2}$ for $T\in M$.)


A countable discrete group $G$ is inner amenable if there is a finitely additive measure $m$ on the subsets of $G \backslash${$e$} with total mass 1 and satisfying $m(gXg^{-1})=mX$ for all $X\subseteq G \backslash${$e$} and all $g \in G.$


I should mention that if the left group von Neumann algebra of an i.c.c. group has property $\Gamma$ then the group is inner amenable, however there exist i.c.c. [inner amenable groups whose group von Neumann algebras don't have $\Gamma$](https://arxiv.org/PS_cache/arxiv/pdf/0909/0909.1485v1.pdf), as recently shown by Stefaan Vaes.



> 
> Given a non-residually finite
> [Baumslag-Solitar group](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-68/issue-3/Some-two-generator-one-relator-non-Hopfian-groups/bams/1183524561.full) $$BS(m,n)
>  = \langle b,s\mid s^{-1}b^ms = b^n\rangle$$ does its group von
> Neumann algebra have property
> $\Gamma$?
> 
> 
> 


It is known that all such groups are inner amenable, and it [recently has been shown that](http://www.math.jussieu.fr/%7Epfima/Documents/Baumslag-Solitar-Groups.pdf) the associated group factors have no Cartan subalgebra, are prime and yet are not solid.



Do Baumslag-Solitar Group von Neumann algebras have Property $\Gamma$?
Yes, it has property $(\Gamma)$. This follows from Stalder's proof of inner amenability plus a fact that the semigroup $\langle T\_m, T\_n \rangle$ admits an approximately invariant subsets having proportional measures. Here, $T\_m$ is the $m$-times map on $[0,1)$, $T\_m x = mx \mod 1$. As Stalder proves, $\sum\_{1\le i,j \le n} z^{m^i n^j}$ is approximately invariant under $T\_m$ and $T\_n$. By a standard procedure, it gives rise a subset $E \subset [0,1)$ which is approximately invariant under $T\_m$ and $T\_n$, i.e., $| T\_l^{-1}(E) \bigtriangleup E | < \epsilon |E|$ for $l=m,n$. Then, Abert--Nikolov's argument (see Chifan--Ioana's paper
arXiv:0802.2353 Lemma 10) allows one to widen $E$. 


