The Haagerup property for locally compact quantum groups

Abstract

AbstractThe Haagerup property for locally compact groups is generalised to the context of locally compact quantum groups, with several equivalent characterisations in terms of the unitary representations and positive-definite functions established. In particular it is shown that a locally compact quantum group 𝔾 has the Haagerup property if and only if its mixing representations are dense in the space of all unitary representations. For discrete 𝔾 we characterise the Haagerup property by the existence of a symmetric proper conditionally negative functional on the dual quantum group