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

Let $M\subset B(\mathcal{H})$ be a von Neumann algebra acting on the Hilbert space $\mathcal{H}$. We prove that $M$ is finite if and only if, for every $x\in M$andforallvectors$\xi,\eta\in\mathcal H$,thecoefficientfunction$u\mapsto \langle uxu^\xi|\eta\rangle$ is weakly almost periodic on the topological group $U_M$ of unitaries in $M$ (equipped with the weak or strong operator topology). The main device is the unique invariant mean on the $C^{\ast }$-algebra $\mathrm{WAP}(U_M)$ of weakly almost periodic functions on $U_M$. Next, we prove that every coefficient function $u\mapsto \langle uxu^\xi|\eta\rangle$isalmostperiodicifandonlyif$M$ 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.