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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Дата:2014
Автор: Alfons Van Daele
Формат: Стаття
Мова:English
Опубліковано: Інститут математики НАН України 2014
Назва видання:Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/146621
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Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати: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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Digital Library of Periodicals of National Academy of Sciences of Ukraine
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spelling irk-123456789-1466212019-02-11T01:23:52Z 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. 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 http://dspace.nbuv.gov.ua/handle/123456789/146621 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description 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
publisher Інститут математики НАН України
publishDate 2014
url http://dspace.nbuv.gov.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.
series Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv AT alfonsvandaele locallycompactquantumgroupsavonneumannalgebraapproach
first_indexed 2023-05-20T17:25:14Z
last_indexed 2023-05-20T17:25:14Z
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