Commutator criteria for strong mixing II. More general and simpler

• Autores: S. Richard, R. Tiedra de Aldecoa
• Localización: Cubo: A Mathematical Journal, ISSN 0716-7776, ISSN-e 0719-0646, Vol. 21, Nº. 1, 2019, págs. 37-48
• Idioma: inglés
• DOI: 10.4067/S0719-06462019000100037
• Enlaces
  • Texto completo
• Resumen
  • español

    Resumen Presentamos un nuevo criterio, basado en métodos de conmutadores, para la propiedad de mezcla fuerte de representaciones unitarias de grupos topológicos dotados de una función de longitud propia. Nuestro resultado generaliza y unifica resultados recientes acerca de la propiedad de mezcla fuerte de flujos discretos {Uᴺ}N∈ℤ y flujos continuos {e-itH}t∈ℝ inducidos por operadores unitarios U y operadores auto adjuntos H en un espacio de Hilbert. Como aplicación, presentamos una demostración corta alternativa (sin usar convoluciones) de la propiedad de mezcla fuerte de la representación regular de grupos localmente compactos σ-compactos.

  • English

    Abstract 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∈ℝ 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.

• Referencias bibliográficas
  • Amrein, W. O.,Boutet de Monvel, A.,Georgescu, V.. (1996). C0-groups, commutator methods andspectral theory of N-body Hamiltonians. BirkhäuserVerlag....
  • Bekka, B.,de la Harpe, P.,Valette, A.. (2008). Kazhdan’s property (T). Cambridge University Press. Cambridge.
  • Bekka, M. B.,Mayer, M.. (2000). Ergodic theory and topological dynamics of group actions on homogeneous spaces. Cambridge University Press....
  • Bergelson, V.,Rosenblatt, J.. (1988). Mixing actions of groups. Illinois J. Math.. 32. 65-80
  • Cecchi, P. A.,Tiedra de Aldecoa, R.. (2016). Furstenberg transformations on cartesian products of infinite-dimensional tori.. Potential Analysis....
  • Cluckers, R.,de Cornulier, Y.,Louvet, N.,Tessera, R.,Valette, A.. (2011). The Howe-Moore property for real and p-adic groups.. Math. Scand.....
  • de Cornulier, Y.,de la Harpe, P.. (2016). Metric geometry of locally compact groups. European Mathematical Society (EMS). Zürich.
  • Fernández, C.,Richard, S.,Tiedra de Aldecoa, R.. (2013). Commutator methods for unitary operators. J. Spectr. Theory. 3. 271
  • Howe, R. E.,Moore, C. C.. (1979). Asymptotic properties of unitary representations. J. Funct. Anal.. 32. 72-96
  • Lubotzky, A.,Mozes, S.. (1991). Harmonic analysis and discrete potential theory. Plenum. New York.
  • Schmidt, K.. (1984). Asymptotic properties of unitary representations and mixing. Proc. London Math. Soc.. 48. 445
  • Tiedra de Aldecoa, R.. (2015). The absolute continuous spectrum of skew products of compact lie groups.. Israel J. Math.. 208. 323
  • Tiedra de Aldecoa, R.. (2012). Spectral analysis of time changes of horocycle flows.. J. Mod. Dyn. 6. 275
  • Tiedra de Aldecoa, R.. (2017). Commutator criteria for strong mixing. Ergodic Theory and Dynam. Systems. 37. 308
  • Tiedra de Aldecoa, R.. (2015). Commutator methods for the spectral analysis of uniquely ergodic dynamical systems. Ergodic Theory Dynam. Systems....
  • Zimmer, R. J.. (1984). Ergodic theory and semisimple groups. Birkhäuser Verlag. Basel.