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The Haagerup property for locally compact quantum groups. (English) Zbl 1343.46068
The Haagerup property of locally compact groups has its origin in Haagerup’s fundamental paper [U. Haagerup, Invent. Math. 50, 279–293 (1979; Zbl 0408.46046)].
In the present paper, the authors undertake a systematic study of the Haagerup property in the setting of locally compact quantum groups. The main result is to show that a locally compact group \(G\) has the Haagerup property if and only if its mixing representations are dense in the space of all unitary representations. In particular, for discrete \(G\), the authors characterize the Haagerup property by the existence of a symmetric proper conditionally negative functional on the dual quantum group \(\widehat{G}\) by the existence of a real proper cocycle on \(G\), and further, if \(G\) is unimodular, then the authors show that the Haagerup property is a von Neumann property of \(G\). This extends results of C. A. Akemann and M. E. Walter [Can. J. Math. 33, 862–871 (1981; Zbl 0437.22004)], M. E. B. Bekka et al. [Lond. Math. Soc. Lect. Note Ser. 227, 1–4 (1995; Zbl 0959.43001)] and P. Jolissaint [J. Oper. Theory 48, No. 3, 549–571 (2002; Zbl 1029.46091); Ergodic Theory Dyn. Syst. 20, No. 2, 483–499 (2000; Zbl 0955.22008); Bull. Belg. Math. Soc. - Simon Stevin 21, No. 2, 263–274 (2014; Zbl 1296.22006)] to the quantum setting and provides a connection to the recent work of M. Brannan [J. Reine Angew. Math. 672, 223–251 (2012; Zbl 1262.46048)]. In the last section of the paper, the authors use these characterizations together with the theory of conditionally free products of states to prove the interesting result that the Haagerup property is preserved under free products of discrete quantum groups.

MSC:
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
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