Almost periodicity was introduced by H. Bohr in the 1920s in the context of functions on the real line. Subsequently, the following generalization has become accepted: a bounded function on a group G is called almost periodic if the set of its translates is relatively compact (in the sup-norm topology). The space of all a.p. functions on G is then an interesting commutative unital C*-algebra, whose spectrum can be regarded as a "compactification" of G.
$L^\infty(G)$ is an example of a Hopf von Neumann algebra, and there are several plausible ways to extend the previous definitions to the world of Hopf von Neumann algebras. In this talk, I will give a brief sketch of some of the classical results, and then discuss a version for Hopf von Neumann algebras that was proposed by Runde, using a modified notion of compactness that may be more appropriate to the operator-space setting.
Extending his results, I shall show that Runde's construction always produces a C*-algebra, and if time permits, I will discuss an unexpected connection with a problem that arose in the study of uniform Roe algebras.