Open Access Open Access  Restricted Access Subscription Access

Haagerup approximation property via bimodules

Rui Okayasu, Narutaka Ozawa, Reiji Tomatsu


The Haagerup approximation property (HAP) is defined for finite von Neumann algebras in such a way that the group von Neumann algebra of a discrete group has the HAP if and only if the group itself has the Haagerup property. The HAP has been studied extensively for finite von Neumann algebras and it was recently generalized to arbitrary von Neumann algebras by Caspers-Skalski and Okayasu-Tomatsu. One of the motivations behind the generalization is the fact that quantum group von Neumann algebras are often infinite even though the Haagerup property has been defined successfully for locally compact quantum groups by Daws-Fima-Skalski-White. In this paper, we fill this gap by proving that the von Neumann algebra of a locally compact quantum group with the Haagerup property has the HAP. This is new even for genuine locally compact groups.

Full Text:



Anantharaman-Delaroche, C., Amenable correspondences and approximation properties for von Neumann algebras, Pacific J. Math. 171 (1995), no. 2, 309–341.

Baaj, S., Skandalis, G., and Vaes, S., Non-semi-regular quantum groups coming from number theory, Comm. Math. Phys. 235 (2003), no. 1, 139–167.

Bannon, J. P. and Fang, J., Some remarks on Haagerup's approximation property, J. Operator Theory 65 (2011), no. 2, 403–417.

Bédos, E., Murphy, G. J., and Tuset, L., Co-amenability of compact quantum groups, J. Geom. Phys. 40 (2001), no. 2, 130–153.

Caspers, M., Okayasu, R., Skalski, A., and Tomatsu, R., Generalisations of the Haagerup approximation property to arbitrary von Neumann algebras, C. R. Math. Acad. Sci. Paris 352 (2014), no. 6, 507–510.

Caspers, M. and Skalski, A., The Haagerup approximation property for von Neumann algebras via quantum Markov semigroups and Dirichlet forms, Comm. Math. Phys. 336 (2015), no. 3, 1637–1664.

Caspers, M. and Skalski, A., The Haagerup property for arbitrary von Neumann algebras, Int. Math. Res. Not. IMRN (2015), no. 19, 9857–9887.

Cherix, P.-A., Cowling, M., Jolissaint, P., Julg, P., and Valette, A., Groups with the Haagerup property, Progress in Mathematics, vol. 197, Birkhäuser Verlag, Basel, 2001.

Choda, M., Group factors of the Haagerup type, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 5, 174–177.

Connes, A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.

Daws, M., Fima, P., Skalski, A., and White, S., The Haagerup property for locally compact quantum groups, J. Reine Angew. Math. 711 (2016), 189–229.

Effros, E. G. and Lance, E. C., Tensor products of operator algebras, Adv. Math. 25 (1977), no. 1, 1–34.

Ghanei, M. R. and Nasr-Isfahani, R., Inner amenability of locally compact quantum groups, Internat. J. Math. 24 (2013), no. 7, 1350058, 17 pp.

Haagerup, U., An example of a nonnuclear $C^ast $-algebra, which has the metric approximation property, Invent. Math. 50 (1978/79), no. 3, 279–293.

Jolissaint, P., Haagerup approximation property for finite von Neumann algebras, J. Operator Theory 48 (2002), no. 3, suppl., 549–571.

Kustermans, J., Locally compact quantum groups in the universal setting, Internat. J. Math. 12 (2001), no. 3, 289–338.

Kustermans, J., Locally compact quantum groups, Quantum independent increment processes. I, Lecture Notes in Math., vol. 1865, Springer, Berlin, 2005, pp. 99--180.

Kustermans, J. and Vaes, S., Locally compact quantum groups, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 6, 837–934.

Lau, A. T. M. and Paterson, A. L. T., Inner amenable locally compact groups, Trans. Amer. Math. Soc. 325 (1991), no. 1, 155–169.

Okayasu, R. and Tomatsu, R., Haagerup approximation property for arbitrary von Neumann algebras, Publ. Res. Inst. Math. Sci. 51 (2015), no. 3, 567–603.

Okayasu, R. and Tomatsu, R., Haagerup approximation property and positive cones associated with a von Neumann algebra, J. Operator Theory 75 (2016), no. 2, 259–288.

Peterson, J. and Sinclair, T., On cocycle superrigidity for Gaussian actions, Ergodic Theory Dynam. Systems 32 (2012), no. 1, 249–272.

Popa, S., Correspondences, INCREST preprint, 1986,

Popa, S., Deformation and rigidity for group actions and von Neumann algebras, International Congress of Mathematicians. Vol. I, Eur. Math. Soc., Zürich, 2007, pp. 445--477.

Takesaki, M., Theory of operator algebras. II, Encyclopaedia of Mathematical Sciences, vol. 125, Operator Algebras and Non-commutative Geometry, 6, Springer-Verlag, Berlin, 2003.

Tomatsu, R., Amenable discrete quantum groups, J. Math. Soc. Japan 58 (2006), no. 4, 949–964.



  • There are currently no refbacks.
This website uses cookies to allow us to see how the site is used. The cookies cannot identify you or any content at your own computer.

ISSN 0025-5521 (print) ISSN 1903-1807 (online)

Hosted by the Royal Danish Library