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

Abstract

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

Section five revised. Some changes in Section eight as a result