Groups with compact open subgroups and multiplier Hopf **-algebras

For a locally compact group GG we look at the group algebras C0(G)C0(G) and Cr*(G), and we let f∈C0(G)f∈C0(G) act on L2(G)L2(G) by the multiplication operator M(f)M(f). We show among other things that the following properties are equivalent:1. GG has a compact open subgroup.2. One of the C*C*-algebras has a dense multiplier Hopf **-subalgebra (which turns out to be unique).3. There are non-zero elements a∈Cr*(G) and f∈C0(G)f∈C0(G) such that aM(f)aM(f) has finite rank.4. There are non-zero elements a∈Cr*(G) and f∈C0(G)f∈C0(G) such that aM(f)=M(f)aaM(f)=M(f)a.If GG is abelian, these properties are equivalent to:5. There is a non-zero continuous function with the property that both ff and f^ have compact support.