Commutator criteria for strong mixing II. More general and simpler
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S. Richard
richard@math.nagoya-u.ac.jp
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R. Tiedra de Aldecoa
rtiedra@mat.puc.cl
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DOI:
https://doi.org/10.4067/S0719-06462019000100037Abstract
We present a new criterion, based on commutator methods, for the strong mixing property of unitary representations of topological groups equipped with a proper length function. Our result generalises and unifies recent results on the strong mixing property of discrete flows {Uᴺ}N∈ℤ and continuous flows {e-itH}t∈℠induced by unitary operators U and self-adjoint operators H in a Hilbert space. As an application, we present a short alternative proof (not using convolutions) of the strong mixing property of the left regular representation of σ-compact locally compact groups.
Keywords
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[2] B. Bekka, P. de la Harpe, and A. Valette. Kazhdan‘s property (T), volume 11 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2008.
[3] M. B. Bekka and M. Mayer. Ergodic theory and topological dynamics of group actions on homogeneous spaces, volume 269 of London Mathematical Society Lecture Note Series. Cam- bridge University Press, Cambridge, 2000.
[4] V. Bergelson and J. Rosenblatt. Mixing actions of groups. Illinois J. Math. 32(1): 65–80, 1988.
[5] P. A. Cecchi and R. Tiedra de Aldecoa. Furstenberg transformations on cartesian products of infinite-dimensional tori. Potential Analysis 44(1): 43–51, 2016.
[6] R. Cluckers, Y. de Cornulier, N. Louvet, R. Tessera, and A. Valette. The Howe-Moore property for real and p-adic groups. Math. Scand. 109(2): 201–224, 2011.
[7] Y. de Cornulier and P. de la Harpe. Metric geometry of locally compact groups, volume 25 of EMS Tracts in Mathematics. European Mathematical Society (EMS), Zu ̈rich, 2016.
[8] C. Ferna Ìndez, S. Richard, and R. Tiedra de Aldecoa. Commutator methods for unitary operators. J. Spectr. Theory 3(3): 271–292, 2013.
[9] R. E. Howe and C. C. Moore. Asymptotic properties of unitary representations. J. Funct. Anal. 32(1): 72–96, 1979.
[10] A. Lubotzky and S. Mozes. Asymptotic properties of unitary representations of tree automorphisms. In Harmonic analysis and discrete potential theory (Frascati, 1991), pages 289–298. Plenum, New York, 1992.
[11] K. Schmidt. Asymptotic properties of unitary representations and mixing. Proc. London Math. Soc. (3) 48(3): 445–460, 1984.
[12] R. Tiedra de Aldecoa. The absolute continuous spectrum of skew products of compact lie groups. Israel J. Math. 208(1): 323–350, 2015.
[13] R. Tiedra de Aldecoa. Spectral analysis of time changes of horocycle flows. J. Mod. Dyn. 6(2): 275–285, 2012.
[14] R. Tiedra de Aldecoa. Commutator criteria for strong mixing. Ergodic Theory and Dynam. Systems 37(1): 308–323, 2017.
[15] R. Tiedra de Aldecoa. Commutator methods for the spectral analysis of uniquely ergodic dynamical systems. Ergodic Theory Dynam. Systems 35(3): 944–967, 2015.
[16] R. J. Zimmer. Ergodic theory and semisimple groups, volume 81 of Monographs in Mathematics. Birkhauser Verlag, Basel, 1984.
[2] B. Bekka, P. de la Harpe, and A. Valette. Kazhdan‘s property (T), volume 11 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2008.
[3] M. B. Bekka and M. Mayer. Ergodic theory and topological dynamics of group actions on homogeneous spaces, volume 269 of London Mathematical Society Lecture Note Series. Cam- bridge University Press, Cambridge, 2000.
[4] V. Bergelson and J. Rosenblatt. Mixing actions of groups. Illinois J. Math. 32(1): 65–80, 1988.
[5] P. A. Cecchi and R. Tiedra de Aldecoa. Furstenberg transformations on cartesian products of infinite-dimensional tori. Potential Analysis 44(1): 43–51, 2016.
[6] R. Cluckers, Y. de Cornulier, N. Louvet, R. Tessera, and A. Valette. The Howe-Moore property for real and p-adic groups. Math. Scand. 109(2): 201–224, 2011.
[7] Y. de Cornulier and P. de la Harpe. Metric geometry of locally compact groups, volume 25 of EMS Tracts in Mathematics. European Mathematical Society (EMS), Zu ̈rich, 2016.
[8] C. Ferna Ìndez, S. Richard, and R. Tiedra de Aldecoa. Commutator methods for unitary operators. J. Spectr. Theory 3(3): 271–292, 2013.
[9] R. E. Howe and C. C. Moore. Asymptotic properties of unitary representations. J. Funct. Anal. 32(1): 72–96, 1979.
[10] A. Lubotzky and S. Mozes. Asymptotic properties of unitary representations of tree automorphisms. In Harmonic analysis and discrete potential theory (Frascati, 1991), pages 289–298. Plenum, New York, 1992.
[11] K. Schmidt. Asymptotic properties of unitary representations and mixing. Proc. London Math. Soc. (3) 48(3): 445–460, 1984.
[12] R. Tiedra de Aldecoa. The absolute continuous spectrum of skew products of compact lie groups. Israel J. Math. 208(1): 323–350, 2015.
[13] R. Tiedra de Aldecoa. Spectral analysis of time changes of horocycle flows. J. Mod. Dyn. 6(2): 275–285, 2012.
[14] R. Tiedra de Aldecoa. Commutator criteria for strong mixing. Ergodic Theory and Dynam. Systems 37(1): 308–323, 2017.
[15] R. Tiedra de Aldecoa. Commutator methods for the spectral analysis of uniquely ergodic dynamical systems. Ergodic Theory Dynam. Systems 35(3): 944–967, 2015.
[16] R. J. Zimmer. Ergodic theory and semisimple groups, volume 81 of Monographs in Mathematics. Birkhauser Verlag, Basel, 1984.
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Published
2019-04-01
How to Cite
[1]
S. . Richard and R. . Tiedra de Aldecoa, “Commutator criteria for strong mixing II. More general and simpler”, CUBO, vol. 21, no. 1, pp. 37–48, Apr. 2019.
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