Bimodules over $\mathop {\rm VN}\nolimits (G)$, harmonic operators and the non-commutative Poisson boundary
Volume 249 / 2019
Abstract
Starting with a left ideal $J$ of $L^1(G)$ we consider its annihilator $J^{\perp }$ in $L^{\infty }(G)$ and the generated $\mathop {\rm VN}\nolimits (G)$-bimodule in $\mathcal {B}(L^2(G))$, $\mathop {\rm Bim}\nolimits (J^{\perp })$. We prove that $ \mathop {\rm Bim}\nolimits (J^{\perp })=(\mathop {\rm Ran}\nolimits J)^{\perp }$ when $G$ is weakly amenable discrete, compact or abelian, where $\mathop {\rm Ran}\nolimits J$ 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 $\mathop {\rm VN}\nolimits (G)$-bimodule generated by the space of jointly harmonic functions. Using this, we give a proof of the following result of Izumi and Ja\-wor\-ski–Neufang: the non-commutative Poisson boundary is isomorphic to the crossed product of the space of harmonic functions by $G$.
