### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 082, 41 pages      math.OA/0602212      https://doi.org/10.3842/SIGMA.2014.082
Contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel

### Locally Compact Quantum Groups. A von Neumann Algebra Approach

Alfons Van Daele
Department of Mathematics, University of Leuven, Celestijnenlaan 200B, B-3001 Heverlee, Belgium

Received February 06, 2014, in final form July 28, 2014; Published online August 05, 2014

Abstract
In this paper, we give an alternative approach to the theory of locally compact quantum groups, as developed by Kustermans and Vaes. We start with a von Neumann algebra and a comultiplication on this von Neumann algebra. We assume that there exist faithful left and right Haar weights. Then we develop the theory within this von Neumann algebra setting. In [Math. Scand. 92 (2003), 68-92] locally compact quantum groups are also studied in the von Neumann algebraic context. This approach is independent of the original $C^*$-algebraic approach in the sense that the earlier results are not used. However, this paper is not really independent because for many proofs, the reader is referred to the original paper where the $C^*$-version is developed. In this paper, we give a completely self-contained approach. Moreover, at various points, we do things differently. We have a different treatment of the antipode. It is similar to the original treatment in [Ann. Sci. École Norm. Sup. (4) 33 (2000), 837-934]. But together with the fact that we work in the von Neumann algebra framework, it allows us to use an idea from [Rev. Roumaine Math. Pures Appl. 21 (1976), 1411-1449] to obtain the uniqueness of the Haar weights in an early stage. We take advantage of this fact when deriving the other main results in the theory. We also give a slightly different approach to duality. Finally, we collect, in a systematic way, several important formulas. In an appendix, we indicate very briefly how the $C^*$-approach and the von Neumann algebra approach eventually yield the same objects. The passage from the von Neumann algebra setting to the $C^*$-algebra setting is more or less standard. For the other direction, we use a new method. It is based on the observation that the Haar weights on the $C^*$-algebra extend to weights on the double dual with central support and that all these supports are the same. Of course, we get the von Neumann algebra by cutting down the double dual with this unique support projection in the center. All together, we see that there are many advantages when we develop the theory of locally compact quantum groups in the von Neumann algebra framework, rather than in the $C^*$-algebra framework. It is not only simpler, the theory of weights on von Neumann algebras is better known and one needs very little to go from the $C^*$-algebras to the von Neumann algebras. Moreover, in many cases when constructing examples, the von Neumann algebra with the coproduct is constructed from the very beginning and the Haar weights are constructed as weights on this von Neumann algebra (using left Hilbert algebra theory). This paper is written in a concise way. In many cases, only indications for the proofs of the results are given. This information should be enough to see that these results are correct. We will give more details in forthcoming paper, which will be expository, aimed at non-specialists. See also [Bull. Kerala Math. Assoc. (2005), 153-177] for an 'expanded' version of the appendix.

Key words: locally compact quantum groups; von Neumann algebras; $C^*$-algebras; left Hilbert algebras.

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References

1. Abe E., Hopf algebras, Cambridge Tracts in Mathematics, Vol. 74, Cambridge University Press, Cambridge - New York, 1980.
2. Baaj S., Skandalis G., Unitaires multiplicatifs et dualité pour les produits croisés de $C^*$-algèbres, Ann. Sci. École Norm. Sup. (4) 26 (1993), 425-488.
3. Delvaux L., Van Daele A., The Drinfel'd double versus the Heisenberg double for an algebraic quantum group, J. Pure Appl. Algebra 190 (2004), 59-84.
4. Enock M., Schwartz J.-M., Kac algebras and duality of locally compact groups, Springer-Verlag, Berlin, 1992.
5. Kirchberg E., Discrete and compact quantum Kac algebras, Lecture at the Conference `Invariance in Operator Algebras' (Copenhagen, August 1992), unpublished.
6. Kustermans J., Vaes S., A simple definition for locally compact quantum groups, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), 871-876.
7. Kustermans J., Vaes S., Locally compact quantum groups, Ann. Sci. École Norm. Sup. (4) 33 (2000), 837-934.
8. Kustermans J., Vaes S., The operator algebra approach to quantum groups, Proc. Natl. Acad. Sci. USA 97 (2000), 547-552.
9. Kustermans J., Vaes S., Locally compact quantum groups in the von Neumann algebraic setting, Math. Scand. 92 (2003), 68-92, math.OA/0005219.
10. Maes A., Van Daele A., Notes on compact quantum groups, Nieuw Arch. Wisk. (4) 16 (1998), 73-112, math.FA/9803122.
11. Maes A., Van Daele A., The multiplicative unitary as a basis for duality, math.OA/0205284.
12. Masuda T., Nakagami Y., A von Neumann algebra framework for the duality of the quantum groups, Publ. Res. Inst. Math. Sci. 30 (1994), 799-850.
13. Masuda T., Nakagami Y., Woronowicz S.L., A $C^\ast$-algebraic framework for quantum groups, Internat. J. Math. 14 (2003), 903-1001, math.QA/0309338.
14. Radford D.E., The order of the antipode of a finite dimensional Hopf algebra is finite, Amer. J. Math. 98 (1976), 333-355.
15. Sołtan P.M., Woronowicz S.L., A remark on manageable multiplicative unitaries, Lett. Math. Phys. 57 (2001), 239-252, math.OA/0604614.
16. Strătilă Ş., Modular theory in operator algebras, Abacus Press, Tunbridge Wells, 1981.
17. Strătilă S., Voiculescu D., Zsidó L., Sur les produits croisés, C. R. Acad. Sci. Paris Sér. A-B 282 (1976), A779-A782.
18. Strătilă Ş., Voiculescu D., Zsidó L., On crossed products. I, Rev. Roumaine Math. Pures Appl. 21 (1976), 1411-1449.
19. Strătilă Ş., Voiculescu D., Zsidó L., On crossed products. II, Rev. Roumaine Math. Pures Appl. 22 (1977), 83-117.
20. Strătilă Ş., Zsidó L., Lectures on von Neumann algebras, Abacus Press, Tunbridge Wells, 1979.
21. Sweedler M.E., Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969.
22. Takesaki M., Tomita's theory of modular Hilbert algebras and its applications, Lecture Notes in Mathematics, Vol. 128, Springer-Verlag, Berlin - New York, 1970.
23. Takesaki M., Theory of operator algebras. I, Springer-Verlag, New York - Heidelberg, 1979.
24. Takesaki M., Theory of operator algebras. II, Springer-Verlag, New York - Heidelberg, 2003.
25. Timmermann T., An invitation to quantum groups and duality. From Hopf algebras to multiplicative unitaries and beyond, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2008.
26. Vaes S., Locally compact quantum groups, Ph.D. Thesis, University of Leuven, 2001.
27. Vaes S., A Radon-Nikodym theorem for von Neumann algebras, J. Operator Theory 46 (2001), 477-489.
28. Vaes S., Van Daele A., Hopf $C^*$-algebras, Proc. London Math. Soc. 82 (2001), 337-384, math.OA/9907030.
29. Vaǐnerman L.Ĭ., Kac G.I., Nonunimodular ring groups and Hopf-von Neumann algebras, Math. USSR Sb. 23 (1974), 185-214.
30. Van Daele A., Dual pairs of Hopf $*$-algebras, Bull. London Math. Soc. 25 (1993), 209-230.
31. Van Daele A., Multiplier Hopf algebras, Trans. Amer. Math. Soc. 342 (1994), 917-932.
32. Van Daele A., Discrete quantum groups, J. Algebra 180 (1996), 431-444.
33. Van Daele A., An algebraic framework for group duality, Adv. Math. 140 (1998), 323-366.
34. Van Daele A., Quantum groups with invariant integrals, Proc. Natl. Acad. Sci. USA 97 (2000), 541-546.
35. Van Daele A., The Haar measure on some locally compact quantum groups, math.OA/0109004.
36. Van Daele A., Locally compact quantum groups: the von Neumann algebra versus the $C^*$-algebra approach, Bull. Kerala Math. Assoc. (2005), 153-177.
37. Van Daele A., Notes on locally compact quantum groups, in preparation.
38. Woronowicz S.L., From multiplicative unitaries to quantum groups, Internat. J. Math. 7 (1996), 127-149.
39. Woronowicz S.L., Compact quantum groups, in Symétries quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998, 845-884.