Banach bimodule associated to an action of a discrete group on a compact space

We associate to an action of a countable discrete group Γ on a compact space X a normal dual Banach N bimodule Z, with separable predual, where N is the group Von Neumann algebra of Γ; and a canonical class of derivations from M to Zo which are inner if and only if there is on X a Γ invariant probability measure.