Locally Compact Quantum Groups. A von Neumann Algebra Approach
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 develo...
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Інститут математики НАН України
2014
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| Zitieren: | Locally Compact Quantum Groups. A von Neumann Algebra Approach / Alfons Van Daele // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 39 назв. — англ. |
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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. |
| format |
Article |
| author |
Alfons Van Daele |
| spellingShingle |
Alfons Van Daele Locally Compact Quantum Groups. A von Neumann Algebra Approach Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Alfons Van Daele |
| author_sort |
Alfons Van Daele |
| title |
Locally Compact Quantum Groups. A von Neumann Algebra Approach |
| title_short |
Locally Compact Quantum Groups. A von Neumann Algebra Approach |
| title_full |
Locally Compact Quantum Groups. A von Neumann Algebra Approach |
| title_fullStr |
Locally Compact Quantum Groups. A von Neumann Algebra Approach |
| title_full_unstemmed |
Locally Compact Quantum Groups. A von Neumann Algebra Approach |
| title_sort |
locally compact quantum groups. a von neumann algebra approach |
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Інститут математики НАН України |
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2014 |
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https://nasplib.isofts.kiev.ua/handle/123456789/146621 |
| citation_txt |
Locally Compact Quantum Groups. A von Neumann Algebra Approach / Alfons Van Daele // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 39 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT alfonsvandaele locallycompactquantumgroupsavonneumannalgebraapproach |
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2025-11-24T16:26:16Z |
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2025-11-24T16:26:16Z |
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nasplib_isofts_kiev_ua-123456789-1466212025-02-23T19:34:59Z Locally Compact Quantum Groups. A von Neumann Algebra Approach Alfons Van Daele 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. This paper is a contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rief fel. The full collection is available at http://www.emis.de/journals/SIGMA/Rieffel.html. Part of this work was done while I was on sabbatical in Trondheim (2002–2003). First of all, I would like to thank my colleague in Leuven, J. Quaegebeur, who did some of my teaching in Leuven and thus made it easier for me to go on sabbatical leave. It is also a pleasure to thank my colleagues of the University in Trondheim, M. Landstad and C. Skau, for their hospitality during my stay and for giving me the opportunity to talk about my work in their seminar. Part of this work was also done while visiting the University of Fukuoka (in 2004) and while visiting the University of Urbana-Champaign (in 2005). I would also like to thank A. Inoue, H. Kurose and Z.-J. Ruan for the hospitality during these (and earlier) visit(s). I am also grateful to my coworkers in Leuven, especially J. Kustermans and S. Vaes, who have always been willing to share their competence in this field. This has been of great importance for my deeper understanding of the theory of locally compact quantum groups. Again, without this, the present paper would not have been written. Finally, I also wish to thank the organizers of the Special Program at the Fields Institute in June 2013 for giving me the opportunity to present this work in a series of lectures. 2014 Article Locally Compact Quantum Groups. A von Neumann Algebra Approach / Alfons Van Daele // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 39 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 26L10; 16L05; 43A99 DOI:10.3842/SIGMA.2014.082 https://nasplib.isofts.kiev.ua/handle/123456789/146621 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України |