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

Abstract: 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 $x\in M$ and for all vectors $\xi,\eta\in\mathcal H$, the coefficient function $u\mapsto \langle uxu<sup>*\xi|\eta\rangle$ 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 $C<sup>*$-algebra $\operatorname{WAP}(U_M)$ of weakly almost periodic functions on $U_M$. Next, we prove that every coefficient function $u\mapsto \langle uxu<sup>*\xi|\eta\rangle$ 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.