Harmonic operators on convolution quantum group algebras

Let $\mathbb{G}$ be a locally compact quantum group and $\mathcal{T}(L^2(\mathbb{G}))$ be the Banach algebra of trace class operators on $L^2(\mathbb{G})$ with the convolution induced by the right fundamental unitary of $\mathbb{G}$. We study the space of harmonic operators ${\tilde{\mathcal{H}}}_{\omega }$ in $\mathcal{B}(L^2(\mathbb{G}))$ associated to a contractive element $\omega \in \mathcal{T}(L^2(\mathbb{G}))$. We characterize the existence of non-zero harmonic operators in $\mathcal{K}(L^2(\mathbb{G}))$ and relate them with some properties of the quantum group $\mathbb{G}$, such as finiteness, amenability and co-amenability.