Harmonic operators on convolution quantum group algebras

Let G be a locally compact quantum group and T(L²( G)) be the Banach algebra of trace class operators on L²( G) with the convolution induced by the right fundamental unitary of G. We study the space of harmonic operators H_ω in B(L²( G)) associated to a contractive element ω∈ T(L²( G)). We characterize the existence of non-zero harmonic operators in K(L²( G)) and relate them with some properties of the quantum group G, such as finiteness, amenability and co-amenability.