Bimodules over vN(G), harmonic operators and  the non-commutative Poisson boundary.

Starting with a left ideal J of L1(G) we consider its annihilator J ⊥ in L ∞ (G) and the generated VN(G)-bimodule in B(L 2 (G)), Bim(J ⊥ ). We prove that Bim(J ⊥ ) = (Ran J) ⊥ when G is weakly amenable discrete, compact or abelian, where Ran J is a suitable saturation of J in the trace class. We define jointly harmonic functions and jointly harmonic operators and show that, for these classes of groups, the space of jointly harmonic operators is the VN(G)-bimodule generated by the space of jointly harmonic functions. Using this, we give a proof of the following result of Izumi and Jaworski – Neufang: the non-commutative Poisson boundary is isomorphic to the crossed product of the space of harmonic functions by G.