Harmonic operators: the dual perspective

The study of harmonic functions on a locally compact group $G$ has recently been transferred to a ``non-commutative'' setting in two different directions: C.-H. Chu and A. T.-M. Lau replaced the algebra $L^\infty(G)$ by the group von Neumann algebra $VN(G)$ and the convolution action of a probability measure $\mu$ on $L^\infty(G)$ by the canonical action of a positive definite function $\sigma$ on $\VN(G)$; on the other hand, W. Jaworski and the first-named author replaced $L^\infty(G)$ by $B(L^2(G))$ to which the convolution action by $\mu$ can be extended in a natural way. We establish a link between both approaches. The action of $\sigma$ on $VN(G)$ can be extended to $B (L^2(G))$. We study the corresponding space $\tilde{H}_\sigma$ of ``$\sigma$-harmonic operators'', i.e., fixed points in $B(L^2(G))$ under the action of $\sigma$. We show, under mild conditions on either $\sigma$ or $G$, that $\tilde{H}_\sigma$ is in fact a von Neumann subalgebra of $B (L^2(G))$. Our investigation of $\tilde{H}_\sigma$ relies, in particular, on a notion of support for an arbitrary operator in $B(L^2(G))$ that extends Eymard's definition for elements of $VN(G)$. Finally, we present an approach to $\tilde{H}_\sigma$ via ideals in $T (L^2(G))$ - where $T(L^2(G))$ denotes the trace class operators on $L^2(G)$, but equipped with a product different from composition -, as it was pioneered for harmonic functions by G. A. Willis.