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

Details Groups with compact open subgroups and multiplier Hopf $^*$-algebras Groups with compact open subgroups and multiplier Hopf $^*$-algebras

2007-01-19

arxiv.org/abs/math/0701525v2

Mathematics

Operator Algebras

23 pages. Section 1 has been shortened and improved. To appear in Expositiones Mathematicae

Scientific paper

For a locally compact group $G$ we look at the group algebras $C_0(G)$ and $C_r^*(G)$, and we let $f\in C_0(G)$ 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 $^*$-subalgebra (which turns out to be unique). 3. There are non-zero elements $a\in C_r^*(G)$ and $f\in C_0(G)$ such that $aM(f)$ has finite rank. 4. There are non-zero elements $a\in C_r^*(G)$ and $f\in C_0(G)$ 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 $\hat f$ have compact support.

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