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

TL;DR

This paper characterizes finiteness of von Neumann algebras via weakly almost periodic functions on their unitary groups and describes conditions under which these functions are almost periodic, revealing structural properties of the algebras.

Contribution

It establishes a new characterization of finite von Neumann algebras using weakly almost periodic functions and describes the structure of algebras with almost periodic coefficient functions.

Findings

M is finite iff all coefficient functions are weakly almost periodic

Coefficient functions are almost periodic iff M is a sum of diffuse abelian and finite-dimensional factors

Diffuse von Neumann algebras have minimally almost periodic unitary groups

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 $u\mapsto \langle uxu^*\xi|\eta\rangle$ 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 $C^*$-algebra $\operatorname{WAP}(U_M)$ of weakly almost periodic functions on $U_M$. Next, we prove that every coefficient function $u\mapsto \langle uxu^*\xi|\eta\rangle$ 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.