Groups with compact open subgroups and multiplier Hopf *-algebras

For a locally compact group G we look at the group algebras C0(G) and , and we let f C0(G) act on L2(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 and f C0(G) such that aM(f) has finite rank. 4. There are non-zero elements and f C0(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 have compact support.