This question concerns alternative characterizations of free group factors. The [ping pong lemma](http://en.wikipedia.org/wiki/Ping-pong_lemma) is a well-known criteria for the freeness of a group. I've often wondered if there is a ping pong like criterion that can be used to determine if a given type $II\_{1}$ factor is a free group factor, e.g. a ping pong-like criterion for the action of the factor on some Hilbert space.



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> Question: Is there a ping pong lemma analogue for group von Neumann algebras?
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Ping Pong and Free Group Factors
I am guessing that the answer is "yes" if you interpret the question in the following way. Let $A\_i$ be some subalgebras of a von Neumann algebra $(M,\tau)$ and assume that there are mutually orthogonal Hilbert subspaces $H\_i$ of $H=L^2(M)$ so that for all $i$, $x (H\ominus H\_i) \subset H\_i$ whenever $x\in A\_i$ with $\tau(x)=0$. 


Let us also assume that $1 \perp \oplus\_i H\_i$ (probably this is not necessary).


Then if $y = x\_1 \dots x\_n$ with $x\_j \in A\_{i(j)}$, $i(1)\neq i(2)$, $i(2)\neq i(3)$, etc. and $\tau (x\_j) = 0$, we have:
$x\_n 1 \in H\_{i(n)}$ since $1\in H\ominus H\_i$;
$x\_{n-1} x\_n 1 \in H\_{i(n-1)}$ since $x\_n 1 \in H\_{i(n)} \subset H\ominus H\_{i(n-1)}$ (because $i(n)\neq i(n-1)$ and so $H(i(n))\perp H(i(n-1))$;
$x\_{n-2} x\_{n-1} x\_n 1 \in H\_{i(n-2)}$ since $x\_{n-1} x\_n 1 \in H\_{i(n-1)}\subset H\ominus H\_{i(n-2)}$, etc. Thus We get that $x\_1\dots x\_n 1 \in H\_{i(1)} \perp 1$, so that $\tau(y)=0$. It follows that $A\_1,\dots,A\_n$ are freely independent.


(Conversely, if $M$ is generated by $A\_1,\dots,A\_n$ and they are free inside of $M$, then $L^2(M) = \mathbb{C}1 \oplus \oplus\_k \oplus\_{j\_1\neq j\_2, j\_2\neq j\_3,\dots} L^2\_0(A\_{j\_1})\otimes \cdots \otimes L^2\_0(A\_{j\_k})$, where $L^2\_0(A\_j) = \{1\}^\perp \cap L^2(A\_j)$. Then you can take $H\_j = \oplus\_k \oplus\_{j\_1\neq j\_2, j\_2\neq j\_3,\dots; j\_1= j} L^2\_0(A\_{j\_1})\otimes \cdots \otimes L^2\_0(A\_{j\_k})$ and then $H\_j$ are orthogonal and 
$H\ominus H\_j$ is taken to $H\_j$ by any $x\in A\_j $ with $\tau(x)=0$).


If you now make some assumption (e.g. that $A\_j$ are finite-dimensional, abelian or hyperfinite) then it follows from Ken Dykema's results (see e.g. his paper on Interpolated free group factors in Duke Math J.) that the von Neumann algebra they generate inside of $M$ is an interpolated free group factor. This is similar to the assumption you have put on the group (since the subgroup generated by a single element in the ping-pong lemma is necessarily abelian).


On the other hand, you raise the much bigger question of whether there exists some criterion that singles out free group factors -- just as the various functional-analytical criteria were shown by Connes to be equivalent to hyperfiniteness. Unfortunately, not much in known in this direction (note that a similar question exists on the ergodic equivalence side of things: is there a functional-analytic way of recognizing treeable actions? Or Bernoulli actions of free groups?)


