# Journal of Operator Theory

Volume 87, Issue 2, Spring 2022 pp. 271-294.

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

**Authors**:
Paul Jolissaint

**Author institution:** Institut de Mathematiques, Universite de Neuchatel, E.-Argand 11,
2000 Neuchatel, Switzerland

**Summary: **Let $M\subset B(\mathcal H)$ be a von Neumann algebra acting on the (separable) Hilbert space $\mathcal H$. We first 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 operator topology). The main device is the unique invariant mean on the $C^*$-algebra $\mathrm{WAP}(U_M)$ of weakly almost periodic functions on $U_M$. Next, we prove that every coefficient function $u\mapsto \langle uxu^*\xi|\eta\rangle$ 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.

**DOI: **http://dx.doi.org/10.7900/jot.2020aug30.2315

**Keywords: **almost periodic functions, minimally almost periodic groups, unitary groups, finite von Neumann algebras, invariant means, conditional expectations

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