Bimodules over vN(G), harmonic operators and the non-commutative Poisson boundary.

Citation:

Anoussis M, Katavolos A, Todorov IG. Bimodules over vN(G), harmonic operators and the non-commutative Poisson boundary. Studia Mathematica [Internet]. 2019;249(2):193-213.

Abstract:

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.

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