  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 $G$.
