There was a small typo in the formula for the product of two projections (n -> m) which has been corrected.

@Lee Mosher: I am aware of exactness of the group being a necessary condition and that this can be expressed as amenability of the natural action on its Stone-Cech compactification. I was not aware of Kaimanovich's work. Thanks for the hint - am I right that 3.15 or 3.17 might be relevant for my situation? After a first scan it looks like it will take me a while to understand the notions involved (and I intend to spend the time). W.r.t. your request I added a realisation of $D$ on a Hilbert space to give a meaning to the projections.

Thanks for the quick reply which apparently does the job. The part about lifting a suitable character from $G/H_1$ to $G$ can even be omitted since $G$ is discrete, so $H_1$ is closed and hence $H_1^{\perp\perp} = H_1$.