The Haagerup property for locally compact quantum groups

TL;DR

This paper extends the concept of the Haagerup property from locally compact groups to quantum groups, establishing equivalent characterisations and demonstrating its preservation under free products.

Contribution

It introduces new characterisations of the Haagerup property for quantum groups and extends classical results to the quantum setting, including stability under free products.

Findings

Equivalent characterisations in terms of unitary representations and positive-definite functions

Haagerup property characterized by symmetric proper conditionally negative functionals for discrete quantum groups

Preservation of the Haagerup property under free products of discrete quantum groups

Abstract

The 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 $\hat{G}$; by the existence of a real proper cocycle on G, and further, if G is also unimodular we show that the Haagerup property is a von Neumann property of G. This extends results of Akemann, Walter, Bekka, Cherix, Valette, and Jolissaint to the quantum setting and provides a connection to the recent work of Brannan. We use these characterisations to show that the Haagerup property is preserved under free products of discrete quantum groups.