The Haagerup property for locally compact quantum groups

Abstract

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 O 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.

One-to-one correspondence between generating functionals and cocycles on quantum groups in presence of symmetry

“…The result turns out to have some important applications: it enables us to show that any quantum Lévy process whose generating functional is S • α-invariant allows the extraction of its maximal Gaussian part and that the Haagerup property for a discrete quantum group is characterised by the existence of a proper (not necessarily real) cocycle. The latter result strengthens Theorem 7.23 in [3].…”

“…Note that this last result, i.e. the S α -invariance for the functional L given by (2.2) in the case of α = id is Proposition 7.22 of [3].…”

“…All these facts can be located in [19], see also [8] (note that we follow rather a convention used for example in [3,9] Finally let us recall that the Hopf * -algebras arising as Pol(G) for a certain compact quantum group G have intrinsic characterization, as CQG-algebras [4].…”

“…= − π(α(a (2) )) * η(S α (a (1) ) * ), η(S α (a (3) )) = η(α(a (1) ) * ), η(S α (a (2) )) . Proof Note first that both expressions on the right hand side of (2.2) coincide due to Lemma 2.6.…”

“…Reality is now understood as a specific interaction with the antipode S of Pol(G). In Kyed's paper this was used to study Kazhdan's Property (T) for discrete quantum groups; recently the same construction was employed to the analysis of the Haagerup property in [3].…”