  We prove a number of results having to do with equipping type-I
$\mathrm{C}^*$-algebras with compact quantum group structures, the two main
ones being that such a compact quantum group is necessarily co-amenable, and
that if the $\mathrm{C}^*$-algebra in question is an extension of a non-zero
finite direct sum of elementary $\mathrm{C}^*$-algebras by a commutative unital
$\mathrm{C}^*$-algebra then it must be finite-dimensional.
