For a locally compact group G we look at the group algebras C0 (G) and Cr* (G), 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 a ∈ Cr* (G) and f ∈ C0 (G) such that aM (f) has finite rank. 4. There are non-zero elements a ∈ Cr* (G) 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 over(f, ^) have compact support. © 2007 Elsevier GmbH. All rights reserved.
Landstad, M. B., & Van Daele, A. (2008). Groups with compact open subgroups and multiplier Hopf *-algebras. Expositiones Mathematicae, 26(3), 197–217. https://doi.org/10.1016/j.exmath.2007.10.004