826 research outputs found
Compactifications of discrete quantum groups
Given a discrete quantum group A we construct a certain Hopf *-algebra AP
which is a unital *-subalgebra of the multiplier algebra of A. The structure
maps for AP are inherited from M(A) and thus the construction yields a
compactification of A which is analogous to the Bohr compactification of a
locally compact group. This algebra has the expected universal property with
respect to homomorphisms from multiplier Hopf algebras of compact type (and is
therefore unique). This provides an easy proof of the fact that for a discrete
quantum group with an infinite dimensional algebra the multiplier algebra is
never a Hopf algebra
Groups with compact open subgroups and multiplier Hopf -algebras
For a locally compact group we look at the group algebras and
, and we let act on by the multiplication
operator . We show among other things that the following properties are
equivalent:
1. has a compact open subgroup.
2. One of the -algebras has a dense multiplier Hopf -subalgebra
(which turns out to be unique).
3. There are non-zero elements and such that
has finite rank.
4. There are non-zero elements and such that
.
If is abelian, these properties are equivalent to:
5. There is a non-zero continuous function with the property that both
and have compact support.Comment: 23 pages. Section 1 has been shortened and improved. To appear in
Expositiones Mathematica
- …