Bimodules over , 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.