Hostname: page-component-76d6cb85b7-xh428 Total loading time: 0 Render date: 2026-07-15T07:03:10.069Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak mixing for locally compact quantum groups

Published online by Cambridge University Press:  28 January 2016

AMI VISELTER
AMI VISELTER*
Affiliation:
Department of Mathematics, University of Haifa, 31905 Haifa, Israel email aviselter@staff.haifa.ac.il
Get access Rights & Permissions [Opens in a new window]

Abstract

We generalize the notion of weakly mixing unitary representations to locally compact quantum groups, introducing suitable extensions of all standard characterizations of weak mixing to this setting. These results are used to complement the non-commutative Jacobs–de Leeuw–Glicksberg splitting theorem of Runde and the author [Ergodic theory for quantum semigroups. J. Lond. Math. Soc. (2) 89(3) (2014), 941–959]. Furthermore, a relation between mixing and weak mixing of state-preserving actions of discrete quantum groups and the properties of certain inclusions of von Neumann algebras, which is known for discrete groups, is demonstrated.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 5 , August 2017 , pp. 1657 - 1680
Copyright
© Cambridge University Press, 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Austin, T.. Lecture Notes on Ergodic Theory. http://cims.nyu.edu/∼tim.Google Scholar
Austin, T., Eisner, T. and Tao, T.. Nonconventional ergodic averages and multiple recurrence for von Neumann dynamical systems. Pacific J. Math. 250(1) (2011), 1–60.CrossRefGoogle Scholar
Bédos, E. and Tuset, L.. Amenability and co-amenability for locally compact quantum groups. Internat. J. Math. 14(8) (2003), 865–884.Google Scholar
Bekka, B., de la Harpe, P. and Valette, A.. Kazhdan’s Property (T) (New Mathematical Monographs, 11) . Cambridge University Press, Cambridge, 2008.CrossRefGoogle Scholar
Bekka, M. E. B. and Valette, A.. Kazhdan’s property (T) and amenable representations. Math. Z. 212(2) (1993), 293–299.Google Scholar
Bergelson, V. and Gorodnik, A.. Weakly mixing group actions: a brief survey and an example. Modern Dynamical Systems and Applications. Cambridge University Press, Cambridge, 2004, pp. 3–25.Google Scholar
Bergelson, V. and Rosenblatt, J.. Mixing actions of groups. Illinois J. Math. 32(1) (1988), 65–80.Google Scholar
Beyers, C., Duvenhage, R. and Ströh, A.. The Szemerédi property in ergodic W ∗ -dynamical systems. J. Operator Theory 64(1) (2010), 35–67.Google Scholar
Cameron, J., Fang, J. and Mukherjee, K.. Mixing subalgebras of finite von Neumann algebras. New York J. Math. 19 (2013), 343–366.CrossRefGoogle Scholar
Chen, X. and Ng, C.-K.. Property T for locally compact quantum groups. Internat. J. Math. 26(3) (2015), 1550024, 13 pp.Google Scholar
Ciorănescu, I. and Zsidó, L.. Analytic generators for one-parameter groups. Tôhoku Math. J. (2) 28(3) (1976), 327–362.Google Scholar
Connes, A. and Weiss, B.. Property T and asymptotically invariant sequences. Israel J. Math. 37(3) (1980), 209–210.Google Scholar
Das, B. and Daws, M.. Quantum Eberlein compactifications and invariant means. Indiana Univ. Math. J., to appear, Preprint, 2014, arXiv:1406.1109v2.Google Scholar
Daws, M.. Remarks on the quantum Bohr compactification. Illinois J. Math. 57(4) (2013), 1131–1171.Google Scholar
Daws, M., Fima, P., Skalski, A. and White, S.. The Haagerup property for locally compact quantum groups. J. Reine Angew. Math., to appear, Preprint, doi:10.1515/crelle-2013-0113.Google Scholar
Daws, M., Kasprzak, P., Skalski, A. and Sołtan, P. M.. Closed quantum subgroups of locally compact quantum groups. Adv. Math. 231(6) (2012), 3473–3501.Google Scholar
de Leeuw, K. and Glicksberg, I.. Applications of almost periodic compactifications. Acta Math. 105 (1961), 63–97.Google Scholar
Dunford, N. and Schwartz, J. T.. Linear Operators. Part II. Spectral Theory (Wiley Classics Library) . John Wiley, New York, 1988.Google Scholar
Duvenhage, R.. Bergelson’s theorem for weakly mixing C ∗ -dynamical systems. Studia Math. 192(3) (2009), 235–257.CrossRefGoogle Scholar
Duvenhage, R.. Ergodicity and mixing of W*-dynamical systems in terms of joinings. Illinois J. Math. 54(2) (2010), 543–566.CrossRefGoogle Scholar
Duvenhage, R.. Relatively independent joinings and subsystems of W∗ -dynamical systems. Studia Math. 209(1) (2012), 21–41.Google Scholar
Duvenhage, R. and Mukhamedov, F.. Relative ergodic properties of C∗ -dynamical systems. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 17(1) (2014), 1450005, 26 pp.Google Scholar
Dye, H. A.. On the ergodic mixing theorem. Trans. Amer. Math. Soc. 118 (1965), 123–130.CrossRefGoogle Scholar
Effros, E. G. and Ruan, Z.-J.. Discrete quantum groups. I. The Haar measure. Internat. J. Math. 5(5) (1994), 681–723.CrossRefGoogle Scholar
Effros, E. G. and Ruan, Z.-J.. Operator Spaces (London Mathematical Society Monographs, New Series, 23) . Clarendon Press, Oxford University Press, New York, 2000.Google Scholar
Enock, M. and Schwartz, J.-M.. Kac Algebras and Duality of Locally Compact Groups. Springer, Berlin, 1992.Google Scholar
Fima, P.. Kazhdan’s property T for discrete quantum groups. Internat. J. Math. 21(1) (2010), 47–65.Google Scholar
Fuglede, B.. A commutativity theorem for normal operators. Proc. Natl. Acad. Sci. USA 36(1) (1950), 35–40.CrossRefGoogle ScholarPubMed
Furstenberg, H.. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. Anal. Math. 31 (1977), 204–256.Google Scholar
Haagerup, U.. The standard form of von Neumann algebras. Math. Scand. 37(2) (1975), 271–283.Google Scholar
Jacobs, K.. Ergodentheorie und fastperiodische Funktionen auf Halbgruppen. Math. Z. 64 (1956), 298–338.Google Scholar
Jolissaint, P. and Stalder, Y.. Strongly singular MASAs and mixing actions in finite von Neumann algebras. Ergod. Th. & Dynam. Sys. 28(6) (2008), 1861–1878.Google Scholar
Koopman, B. O. and von Neumann, J.. Dynamical systems of continuous spectra. Proc. Natl. Acad. Sci. USA 18(3) (1932), 255–263.CrossRefGoogle ScholarPubMed
Krengel, U.. Ergodic Theorems (de Gruyter Studies in Mathematics, 6) . Walter de Gruyter, Berlin, 1985.Google Scholar
Kustermans, J.. Locally compact quantum groups in the universal setting. Internat. J. Math. 12(3) (2001), 289–338.Google Scholar
Kustermans, J. and Vaes, S.. Locally compact quantum groups. Ann. Sci. Éc. Norm. Supér. (4) 33(6) (2000), 837–934.Google Scholar
Kustermans, J. and Vaes, S.. Locally compact quantum groups in the von Neumann algebraic setting. Math. Scand. 92(1) (2003), 68–92.Google Scholar
Kyed, D.. A cohomological description of property (T) for quantum groups. J. Funct. Anal. 261(6) (2011), 1469–1493.Google Scholar
Kyed, D. and Sołtan, P. M.. Property (T) and exotic quantum group norms. J. Noncommut. Geom. 6(4) (2012), 773–800.Google Scholar
Maes, A. and Van Daele, A.. Notes on compact quantum groups. Nieuw Arch. Wiskd. (4) 16(1–2) (1998), 73–112.Google Scholar
Masuda, T., Nakagami, Y. and Woronowicz, S. L.. A C ∗ -algebraic framework for quantum groups. Internat. J. Math. 14(9) (2003), 903–1001.Google Scholar
Meyer, R., Roy, S. and Woronowicz, S. L.. Homomorphisms of quantum groups. Münster J. Math. 5 (2012), 1–24.Google Scholar
Niculescu, C. P., Ströh, A. and Zsidó, L.. Noncommutative extensions of classical and multiple recurrence theorems. J. Operator Theory 50(1) (2003), 3–52.Google Scholar
Okayasu, R., Ozawa, N. and Tomatsu, R.. Haagerup approximation property via bimodules, Preprint, 2015, arXiv:1501.06293v2.Google Scholar
Peterson, J.. Lecture Notes on Ergodic Theory. http://www.math.vanderbilt.edu/∼peters10/teaching/Spring2011/math390Blecturenotes.html.Google Scholar
Peterson, J. and Sinclair, T.. On cocycle superrigidity for Gaussian actions. Ergod. Th. & Dynam. Sys. 32(1) (2012), 249–272.Google Scholar
Popa, S.. On a class of type II1 factors with Betti numbers invariants. Ann. of Math. (2) 163(3) (2006), 809–899.Google Scholar
Popa, S.. Some rigidity results for non-commutative Bernoulli shifts. J. Funct. Anal. 230(2) (2006), 273–328.Google Scholar
Popa, S.. Strong rigidity of II1 factors arising from malleable actions of w-rigid groups. I. Invent. Math. 165(2) (2006), 369–408.Google Scholar
Popa, S.. Strong rigidity of II1 factors arising from malleable actions of w-rigid groups. II. Invent. Math. 165(2) (2006), 409–451.Google Scholar
Popa, S.. Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups. Invent. Math. 170(2) (2007), 243–295.Google Scholar
Popa, S. and Vaes, S.. Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups. Adv. Math. 217(2) (2008), 833–872.Google Scholar
Robertson, G., Sinclair, A. M. and Smith, R. R.. Strong singularity for subalgebras of finite factors. Internat. J. Math. 14(3) (2003), 235–258.Google Scholar
Runde, V.. Characterizations of compact and discrete quantum groups through second duals. J. Operator Theory 60(2) (2008), 415–428.Google Scholar
Runde, V. and Viselter, A.. Ergodic theory for quantum semigroups. J. Lond. Math. Soc. (2) 89(3) (2014), 941–959.CrossRefGoogle Scholar
Sinclair, A. M. and Smith, R. R.. Strongly singular masas in type II1 factors. Geom. Funct. Anal. 12(1) (2002), 199–216.Google Scholar
Sinclair, A. M., Smith, R. R., White, S. A. and Wiggins, A.. Strong singularity of singular masas in II1 factors. Illinois J. Math. 51(4) (2007), 1077–1084.Google Scholar
Sołtan, P. M.. Quantum Bohr compactification. Illinois J. Math. 49(4) (2005), 1245–1270.CrossRefGoogle Scholar
Strătilă, Ş.. Modular Theory in Operator Algebras. Abacus Press, Tunbridge Wells, UK, 1981.Google Scholar
Takesaki, M.. Theory of Operator Algebras. II (Encyclopaedia of Mathematical Sciences, 125) . Springer, Berlin, 2003.Google Scholar
Vaes, S.. The unitary implementation of a locally compact quantum group action. J. Funct. Anal. 180(2) (2001), 426–480.Google Scholar
Vaes, S.. Strictly outer actions of groups and quantum groups. J. Reine Angew. Math. 578 (2005), 147–184.Google Scholar
Van Daele, A.. Discrete quantum groups. J. Algebra 180(2) (1996), 431–444.Google Scholar
Woronowicz, S. L.. A remark on compact matrix quantum groups. Lett. Math. Phys. 21(1) (1991), 35–39.Google Scholar
Woronowicz, S. L.. Compact quantum groups. Symétries Quantiques (Les Houches, 1995). North-Holland, Amsterdam, 1998, pp. 845–884.Google Scholar
Zsidó, L.. Spectral and ergodic properties of the analytic generators. J. Approx. Theory 20(1) (1977), 77–138.Google Scholar
Zsidó, L.. Splitting noncommutative dynamical systems in almost periodic and weakly mixing parts. Presented at Int. Conf. on Operator Theory (Timişoara, Romania, 2012).Google Scholar