You posted the link to the question, and my answer is there; it was the one accepted. The techniques are borrowed from the Complex case to use harmonic functions instead, which is what is effectively used in the Complex case anyway.

@Andras Batkai: I don't see "that this connection was maid by von Neumann" in Dieudonne. Did I miss a statement of that kind? It would not surprise me that von Neumann put the pieces together, but I didn't see such a reference.