## Toeplitz operator used to study the free group factors?

Note that when we construct the 'group von Neumann algebra'$vN(\gamma)$ using a discrete group $\gamma$(especially,the free group on n generators,$n \geq 2$),the elements of $vN(\gamma)$ have matrices(w.r.t, the basis $\epsilon_\gamma$) which are constant along the 'diagonals':{($\gamma,\tau$):$\gamma \tao^{-1}$ is constant}, which is a Toeplitz matrix.

Recall that a Toeplitz operator can also be represented as a Toeplitz matrix, so I want to know:

Are there known results linked to the free group factors obtained by applying techniques from studying the Toeplitz operator?

-
 The "Toeplitz" operators (which I would just call convolution operators) on a general discrete group $\Gamma$ can be much more complicated than for the case $\Gamma={\mathbb Z}$, which is the "classical" doubly-infinite Toeplitz example you mention. – Yemon Choi Jun 25 2011 at 20:37 Speaking subjectively, my instinct is that your question is really too optimistic. Or, "techniques from studying the Toeplitz operator" will turn out to be as hard as the original questions concerning free group factors. However, I am not a specialist in operator algebras nor operator theory, so perhaps someone will provide evidence contrary to my first thoughts. – Yemon Choi Jun 25 2011 at 20:40