Almost and weakly almost periodic functions on the unitary groups of von Neumann algebras

Let M⊂B(H) be a von Neumann algebra acting on the (separable) Hilbert space H. We first prove that M is finite if and only if, for every x∈M and for all vectors ξ,η∈H, the coefficient function u↦⟨uxu∗ξ|η⟩ is weakly almost periodic on the topological group UM of unitaries in M (equipped with the weak operator topology). The main device is the unique invariant mean on the C∗-algebra WAP(UM) of weakly almost periodic functions on UM. Next, we prove that every coefficient function u↦⟨uxu∗ξ|η⟩ is almost periodic if and only if M is a direct sum of a diffuse, abelian von Neumann algebra and finite-dimensional factors. Incidentally, we prove that if M is a diffuse von Neumann algebra, then its unitary group is minimally almost periodic.