  For a locally compact group $G$ we look at the group algebras $C_0(G)$ and
$C_r^*(G)$, and we let $f\in C_0(G)$ act on $L^2(G)$ by the multiplication
operator $M(f)$. We show among other things that the following properties are
equivalent:
  1. $G$ has a compact open subgroup.
  2. One of the $C^*$-algebras has a dense multiplier Hopf $^*$-subalgebra
(which turns out to be unique).
  3. There are non-zero elements $a\in C_r^*(G)$ and $f\in C_0(G)$ such that
$aM(f)$ has finite rank.
  4. There are non-zero elements $a\in C_r^*(G)$ and $f\in C_0(G)$ such that
$aM(f)=M(f)a$.
  If $G$ is abelian, these properties are equivalent to:
  5. There is a non-zero continuous function with the property that both $f$
and $\hat f$ have compact support.
