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Poisson boundary of the discrete quantum group

Part of: Markov processes Selfadjoint operator algebras Groups and algebras in quantum theory

Published online by Cambridge University Press:  08 June 2010

Stefaan Vaes  and
Nikolas Vander Vennet
Stefaan Vaes
Affiliation:
Department of Mathematics, K.U. Leuven, Celestijnenlaan 200B, B–3001 Leuven, Belgium (email: [email protected])
Nikolas Vander Vennet
Affiliation:
Department of Mathematics, K.U. Leuven, Celestijnenlaan 200B, B–3001 Leuven, Belgium (email: [email protected])
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Abstract

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We identify the Poisson boundary of the dual of the universal compact quantum group Au(F) with a measurable field of ITPFI (infinite tensor product of finite type I) factors.

Keywords

discrete quantum groupcompact quantum grouprandom walkPoisson boundaryMartin boundaryfree quantum groupsolid II1 factorboundary action

MSC classification

Secondary: 46L53: Noncommutative probability and statistics 46L54: Free probability and free operator algebras 60J50: Boundary theory 81R50: Quantum groups and related algebraic methods
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 4 , July 2010 , pp. 1073 - 1095
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Banica, T., Théorie des représentations du groupe quantique compact libre O(n), C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), 241–244.Google Scholar
[2]Banica, T., Le groupe quantique compact libre U(n), Commun. Math. Phys. 190 (1997), 143–172.CrossRefGoogle Scholar
[3]Bichon, J., De Rijdt, A. and Vaes, S., Ergodic coactions with large quantum multiplicity and monoidal equivalence of quantum groups, Comm. Math. Phys 262 (2006), 703–728.CrossRefGoogle Scholar
[4]De Rijdt, A. and Vander Vennet, N., Actions of monoidally equivalent quantum groups and applications to probabilistic boundaries, Ann. Inst. Fourier (Grenoble) 60 (2010), 169–216.CrossRefGoogle Scholar
[5]Izumi, M., Non-commutative Poisson boundaries and compact quantum group actions, Adv. Math. 169 (2002), 1–57.CrossRefGoogle Scholar
[6]Izumi, M., Neshveyev, S. and Tuset, L., Poisson boundary of the dual of SU q(n), Comm. Math. Phys. 262 (2006), 505–531.CrossRefGoogle Scholar
[7]Neshveyev, S. and Tuset, L., The Martin boundary of a discrete quantum group, J. Reine Angew. Math. 568 (2004), 23–70.Google Scholar
[8]Ozawa, N., Solid von Neumann algebras, Acta Math. 192 (2004), 111–117.CrossRefGoogle Scholar
[9]Picardello, M. and Woess, W., Martin boundaries of random walks: ends of trees and groups, Trans. Amer. Math. Soc. 302 (1987), 185–205.CrossRefGoogle Scholar
[10]Tomatsu, R., A characterization of right coideals of quotient type and its application to classification of Poisson boundaries, Comm. Math. Phys. 275 (2007), 271–296.CrossRefGoogle Scholar
[11]Vaes, S. and Vergnioux, R., The boundary of universal discrete quantum groups, exactness, and factoriality, Duke Math. J. 140 (2007), 35–84.CrossRefGoogle Scholar
[12]Vaes, S. and Vander Vennet, N., Identification of the Poisson and Martin boundaries of orthogonal discrete quantum groups, J. Inst. Math. Jussieu 7 (2008), 391–412.CrossRefGoogle Scholar
[13]Van Daele, A., Discrete quantum groups, J. Algebra 180 (1996), 431–444.CrossRefGoogle Scholar
[14]Van Daele, A. and Wang, S., Universal quantum groups, Internat. J. Math. 7 (1996), 255–263.CrossRefGoogle Scholar
[15]Vander Vennet, N., Probabilistic boundaries of discrete quantum groups. PhD thesis, K.U. Leuven, (2008), available at http://hdl.handle.net/1979/1864.Google Scholar
[16]Woess, W., Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics, vol. 138 (Cambridge University Press, Cambridge, 2000).CrossRefGoogle Scholar
[17]Woronowicz, S. L., Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613–665.CrossRefGoogle Scholar
[18]Woronowicz, S. L., Compact quantum groups, in Symétries quantiques (Les Houches, 1995) (North-Holland, Amsterdam, 1998), 845–884.Google Scholar