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Multiplier Hopf algebras and duality

A. van Daele (1997)

Banach Center Publications

We define a category containing the discrete quantum groups (and hence the discrete groups and the duals of compact groups) and the compact quantum groups (and hence the compact groups and the duals of discrete groups). The dual of an object can be defined within the same category and we have a biduality theorem. This theory extends the duality between compact quantum groups and discrete quantum groups (and hence the one between compact abelian groups and discrete abelian groups). The objects in...

On a translation property of positive definite functions

Lars Omlor, Michael Leinert (2010)

Banach Center Publications

If G is a locally compact group with a compact invariant neighbourhood of the identity e, the following property (*) holds: For every continuous positive definite function h≥ 0 with compact support there is a constant C h > 0 such that ∫ L x h · g ≤ C h ∫ h g for every continuous positive definite g≥0, where L x is left translation by x. In [L], property (*) was stated, but the above inequality was proved for special h only. That “for one h” implies “for all h” seemed obvious, but turned out not to be obvious at all. We fill...

The dual space of precompact groups

M. Ferrer, S. Hernández, V. Uspenskij (2013)

Commentationes Mathematicae Universitatis Carolinae

For any topological group G the dual object G ^ is defined as the set of equivalence classes of irreducible unitary representations of G equipped with the Fell topology. If G is compact, G ^ is discrete. In an earlier paper we proved that G ^ is discrete for every metrizable precompact group, i.e. a dense subgroup of a compact metrizable group. We generalize this result to the case when G is an almost metrizable precompact group.

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