Bimodules over ${rm VN}(G)$, harmonic operators and the non-commutative Poisson boundary

Starting with a left ideal $J$ of $L^1(G)$ we consider its annihilator $J^{perp}$ in $L^{infty}(G)$ and the generated ${rm VN}(G)$-bimodule in $mathcal{B}(L^2(G))$, ${rm Bim}(J^{perp})$. We prove that ${rm Bim}(J^{perp})=({rm Ran} J)^{perp}$ when $G$ is weakly amenable discrete, compact or abelian, where ${rm Ran} J$ 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 ${rm VN}(G)$-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 $G$.