O ct 2 00 7 GROUPS WITH COMPACT OPEN SUBGROUPS AND MULTIPLIER HOPF ∗ - ALGEBRAS MAGNUS

For a locally compact group G we look at the group algebras C 0 (G) and C * r (G), and we let f ∈ 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 *-subalg-ebra (which turns out to be unique). 3. There are non-zero elements a ∈ C * r (G) and f ∈ C 0 (G) such that aM (f) has finite rank. 4. There are non-zero elements a ∈ C * r (G) and f ∈ 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 f have compact support.