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

TL;DR

This paper explores bimodules over group von Neumann algebras, characterizes harmonic operators, and establishes a connection between the non-commutative Poisson boundary and harmonic functions for specific classes of groups.

Contribution

It provides a new characterization of bimodules generated by harmonic functions and proves the isomorphism between the non-commutative Poisson boundary and a crossed product for certain groups.

Findings

Bimodule equality for weakly amenable groups

Harmonic operators generated by harmonic functions

Non-commutative Poisson boundary is a crossed product

Abstract

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