Starting with a left ideal J of L1(G) we consider its annihilator J ⊥ in L ∞ (G) and the generated VN(G)-bimodule in B(L 2 (G)), Bim(J ⊥ ). We prove that Bim(J ⊥ ) = (Ran J) ⊥ when G is weakly amenable discrete, compact or abelian, where 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 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.
Katavolos A. von Neumann algebras and unbounded operators (in Greek). In: Anoussis M, Kotsakis S, Hatzisavvas N Operator Algebras and Quantum Mechanics (Proceedings of the third Summer School in Analysis, Geometry and Mathematican Physics). Thessaloniki : Ziti Publishers; 1997.