# American Institute of Mathematical Sciences

2017, 11: 99-123. doi: 10.3934/jmd.2017005

## On spectra of Koopman, groupoid and quasi-regular representations

 1 Department of Mathematics, University of Toronto, Room 6290, 40 St. George Street, Toronto, ON M5S 2E4, Canada 2 Department of Mathematics, MS 3368, Texas A&M University, College Station, TX 77843-3368, USA

Received  May 31, 2016 Revised  November 05, 2016 Published  January 2017

Fund Project: Both authors were supported by the Swiss National Science Foundation.
RG:Supported by NSF grant DMS-1207699 and NSA grant H98230-15-1-0328

In this paper we investigate relations between Koopman, groupoid and quasi-regular representations of countable groups. We show that for an ergodic measure class preserving action of a countable group G on a standard Borel space the associated groupoid and quasi-regular representations are weakly equivalent and weakly contained in the Koopman representation. Moreover, if the action is hyperfinite then the Koopman representation is weakly equivalent to the groupoid. As a corollary of our results we obtain a continuum of pairwise disjoint pairwise equivalent irreducible representations of weakly branch groups. As an illustration we calculate spectra of regular, Koopman and groupoid representations associated to the action of the 2-group of intermediate growth constructed by the second author in 1980.

Citation: Artem Dudko, Rostislav Grigorchuk. On spectra of Koopman, groupoid and quasi-regular representations. Journal of Modern Dynamics, 2017, 11: 99-123. doi: 10.3934/jmd.2017005
##### References:
 [1] S. Adams, G. Elliot and T. Giordano, Amenable actions of groups, Trans. Amer. Math. Soc., 344 (1994), 803-822. doi: 10.1090/S0002-9947-1994-1250814-5. Google Scholar [2] C. Anantharaman-Delaroche, On spectral characterization of amenability, Israel Journal of Mathematics, 137 (2003), pp. 1-33. doi: 10.1007/BF02785954. Google Scholar [3] U. Bader and R. Muchnik, Boundary unitary representations – irreducibility and rigidity, Journal of Modern Dynamics, 5 (2011), pp. 49-69. doi: 10.3934/jmd.2011.5.49. Google Scholar [4] L. Bartholdi and R. I. Grigorchuk, On the Spectrum of Hecke Type Operators Related to Some Fractal Groups, Tr. Mat. Inst. im. V. A. Steklova, Ross. Akad. Nauk 231 (2000), 5-45; Proc. Steklov Inst. Math. 231 (2000), 1-41. Google Scholar [5] L. Bartholdi, R. I. Grigorchuk, and Z. Šunić, Branch Groups, Handbook of Algebra (NorthHolland, Amsterdam, 2003), Vol. 3, pp. 989-1112. Google Scholar [6] M. E. B. Bekka and M. Cowling, Some irreducible unitary representation of G(K) for a simple algebraic group G over an algebraic number field K, Math. Z., 241 (2002), 731-741. doi: 10.1007/s00209-002-0442-6. Google Scholar [7] B. Bekka, P. de la Harpe and A. Valette, Kazhdan Property (T), Cambridge University Press 2008. Google Scholar [8] O. Brattelli and W. Robinson, Operator algebras and quantum statistical mechanics. C*-and W *-algebras. Symmetry Groups. Decomposition of States. Texts and Monographs in Physics. Berlin-Heidelberg-New York, Springer-Verlag, 1987. Google Scholar [9] R.V. Chacon and N.A. Friedman, Approximation and invariant measures, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 3 (1965), 286-295. doi: 10.1007/BF00535780. Google Scholar [10] T. Chow, A spectral theory for direct integrals of operators, Math. Ann., 188 (1970), pp. 285-303. doi: 10.1007/BF01431463. Google Scholar [11] M. Cowling and T. Steger, The irreducibility of restrictions of unitary representations to lattices, J. Reine Angew. Math., 420 (1991), 85-98. Google Scholar [12] J. Dixmier, Les C*-algèbres et leurs représentations, Gauthier-Villars 1969. Google Scholar [13] A. Dudko, R. Grigorchuk, On diagonal actions of branch groups and the corresponding characters, arxiv: 1412.5476.Google Scholar [14] A. Dudko, R. Grigorchuk, On irreducibility and disjointness of Koopman and quasi-regular representations of weakly branch groups, Modern Theory of Dynamical Systems: A Tribute to Dmitry Victorovich Anosov, AMS Cont. Math. Ser. (2016), to appear.Google Scholar [15] A. Dudko and K. Medynets, On Characters of Inductive Limits of Symmetric Groups, Journal of Functional Analysis,, 264 (2013), 1565-1598. doi: 10.1016/j.jfa.2013.01.013. Google Scholar [16] J. Feldman and C. Moore, Ergodic equivalence relations. cohomology, and von Neumann algebras.Ⅰ, Trans. Amer. Math. Soc., 234 (1977), 289-324. doi: 10.1090/S0002-9947-1977-0578656-4. Google Scholar [17] J. Feldman and C. Moore, Ergodic equivalence relations. cohomology, and von Neumann algebras. Ⅱ, Trans. Amer. Math. Soc., 234 (1977), 325-359. doi: 10.1090/S0002-9947-1977-0578730-2. Google Scholar [18] A. Figá-Talamanca and M. A. Picardello, Harmonic analysis on free groups, Lecture Notes in Pure and Applied Mathematics, vol. 87, Marcel Dekker Inc., New York, (1983). Google Scholar [19] A. Figá-Talamanca, T. Steger, Harmonic analysis for anisotropic random walks on homogeneous trees, Mem. Amer. Math. Soc. 110 (1994), no. 531, ⅶ+68. Google Scholar [20] E. Glasner, Ergodic theory via joinings, Math. Surv. and Mon., 101, Amer. Math. Soc. (2003). Google Scholar [21] R. I. Grigorchuk, Burnside's Problem on Periodic Groups, Funkts. Anal. Prilozh. 14 (1), 53-54 (1980)[Funct. Anal. Appl. 14, 41-43 (1980)]. Google Scholar [22] R. I. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 48 (1984), 939-985. Google Scholar [23] R. I. Grigorchuk, Degrees of growth of p-groups and torsion-free groups (Russian) Mat. Sb. (N. S. ) 126 (168) (1985),194-214. Google Scholar [24] R. I. Grigorchuk, Just Infinite Branch Groups, New Horizons in Pro-p Groups (Birkhauser, Boston, MA, 2000), Prog. Math. 184, pp. 121-179. Google Scholar [25] R. I. Grigorchuk, Some topics in the dynamics of group actions on rooted trees, Proceedings of the Steklov Institute of Mathematics, 273, Issue 1 (2011), pp. 64-175. Google Scholar [26] R. I. Grigorchuk, D. Lenz and T. Smirnova-Nagnibeda, Spectra of Schreier graphs of Grigorchuk's group and Schroedinger operators with aperiodic order, arXiv: 1412.6822.Google Scholar [27] R. I. Grigorchuk and V. Nekrashevich, Amenable actions of non-amenable groups, J. of Math. Sci., 140 (2007), pp. 391-397. Google Scholar [28] R. Grigorchuk, V. Nekrashevich, and Z. Sunic, From self-similar groups to self-similar sets and spectra, Fractal Geometry and Stochastics V, 70, series Progress in Probability, pp. 175-207. doi: 10.1007/978-3-319-18660-3_11. Google Scholar [29] R. I. Grigorchuk, V. V. Nekrashevich, and V. I. Sushchanskii, Automata, Dynamical Systems, and Groups, Tr. Mat. Inst. im. V. A. Steklova, Ross. Akad. Nauk 231,134-214 (2000)[Proc. Steklov Inst. Math. 231,128-203 (2000)]. Google Scholar [30] R. I. Grigorchuk and Ya. Krylyuk, The spectral measure of the Markov operator related to 3-generated 2-group of intermediate growth and its Jacobi parameters, Algebra Discrete Math., 13 (2012), pp. 237-272. Google Scholar [31] P. de la Harpe, On simplicity of reduced C*-Ualgebras of groups, Bull. London Math. Soc., 39 (2007), pp. 1-26. doi: 10.1112/blms/bdl014. Google Scholar [32] N. Higson and G. Kasparov, Operator K-theory for groups which act properly and isometrically on Hilbert space, Electron. Res. Announc. Amer. Math. Soc., 3 (1997), 131-142. doi: 10.1090/S1079-6762-97-00038-3. Google Scholar [33] R. Kadison and J. Ringrose, Fundamentals of the theory of operator algebras. Vol. Ⅰ. Elementary theory. Pure and Applied Mathematics, 100. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. Google Scholar [34] R. Kadison and J. Ringrose, Fundamentals of the theory of operator algebras. Vol. Ⅱ. Advanced theory. Pure and Applied Mathematics, 100. Academic Press, Inc., Orlando, FL, 1986. Google Scholar [35] I. Lysenok, A set of defining relations for the Grigorchuk group, Mat. Zametki, 38 (1985), pp. 503U516.. Google Scholar [36] S. Kerov and A. Vershik, Characters and factor representations of the infinite symmetric group, Soviet Math. Dokl., 23 (1981), 389-392. Google Scholar [37] H. Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc., 92 (1959), 336-354. doi: 10.1090/S0002-9947-1959-0109367-6. Google Scholar [38] G. Kuhn, Amenable actions and weak containment of certain representations of discrete groups, Proc. Amer. Math. Soc., 122 (1994), 751-757. doi: 10.1090/S0002-9939-1994-1209424-3. Google Scholar [39] G. Kuhn and T. Steger, More irreducible boundary representations of free groups, Duke Math. J., 82, no. 2 (1996), 381-435. doi: 10.1215/S0012-7094-96-08218-6. Google Scholar [40] G. W. Mackey, The Theory of Unitary Group Representations, Univ. Chicago Press, Chicago, 1976, Chicago Lect. Math. Google Scholar [41] M. Naimark, Normed rings, Noordhoff, Groningen, 1959, xvi+560pp. Google Scholar [42] V. Nekrashevych, Self-similar Groups, Am. Math. Soc., Providence, RI, 2005, Math. Surv. Monogr. 117. Google Scholar [43] M. Pichot, Sur la théorie spectrale des relations d'équivalence mesurées, Journal of the Inst. of Math. Jussieu, 3, 6 (2007), 453-500. doi: 10.1017/S1474748007000096. Google Scholar [44] D. Powers, Simplicity of the C*-algebra associated with the free group on two generators, Duke Math. J., 42 (1975), pp. 151-156. doi: 10.1215/S0012-7094-75-04213-1. Google Scholar [45] V. A. Rokhlin, On the basic ideas of measure theory, Mat. Sb. 25 (1) (1949), 107-150. Engl. transl., On the fundamental ideas of measure theory, (Am. Math. Soc., Providence, RI, 1952), AMS Transl., No. 71, 55pp. Google Scholar [46] V. A. Rokhlin, Selected topics from the metric theory of dynamical systems, (Russian) Us-pekhi Mat. Nauk, 4 (1949), 57–128 Google Scholar [47] H. L. Skudlarek, Die unzerlegbaren Charaktere einiger diskreter Gruppen, Math. Ann., 223 (1976), 213-231. doi: 10.1007/BF01360954. Google Scholar [48] M. Takesaki, Theory of Operator Algebras. I, Encyclopedia of Mathematical Sciences, 124, Operator Algebras and Non-commutative Geometry, 5, Springer-Verlag, Berlin, 2002. Google Scholar [49] M. Takesaki, Theory of Operator Algebras. III, Encyclopedia of Mathematical Sciences, 127, Operator Algebras and Non-commutative Geometry, 8, Springer-Verlag, Berlin, 2003. Google Scholar [50] R.J. Zimmer, Hyperfinite factors and amenable ergodic actions, Inv. Math., 41 (1977), 23-31. doi: 10.1007/BF01390162. Google Scholar

show all references

##### References:
 [1] S. Adams, G. Elliot and T. Giordano, Amenable actions of groups, Trans. Amer. Math. Soc., 344 (1994), 803-822. doi: 10.1090/S0002-9947-1994-1250814-5. Google Scholar [2] C. Anantharaman-Delaroche, On spectral characterization of amenability, Israel Journal of Mathematics, 137 (2003), pp. 1-33. doi: 10.1007/BF02785954. Google Scholar [3] U. Bader and R. Muchnik, Boundary unitary representations – irreducibility and rigidity, Journal of Modern Dynamics, 5 (2011), pp. 49-69. doi: 10.3934/jmd.2011.5.49. Google Scholar [4] L. Bartholdi and R. I. Grigorchuk, On the Spectrum of Hecke Type Operators Related to Some Fractal Groups, Tr. Mat. Inst. im. V. A. Steklova, Ross. Akad. Nauk 231 (2000), 5-45; Proc. Steklov Inst. Math. 231 (2000), 1-41. Google Scholar [5] L. Bartholdi, R. I. Grigorchuk, and Z. Šunić, Branch Groups, Handbook of Algebra (NorthHolland, Amsterdam, 2003), Vol. 3, pp. 989-1112. Google Scholar [6] M. E. B. Bekka and M. Cowling, Some irreducible unitary representation of G(K) for a simple algebraic group G over an algebraic number field K, Math. Z., 241 (2002), 731-741. doi: 10.1007/s00209-002-0442-6. Google Scholar [7] B. Bekka, P. de la Harpe and A. Valette, Kazhdan Property (T), Cambridge University Press 2008. Google Scholar [8] O. Brattelli and W. Robinson, Operator algebras and quantum statistical mechanics. C*-and W *-algebras. Symmetry Groups. Decomposition of States. Texts and Monographs in Physics. Berlin-Heidelberg-New York, Springer-Verlag, 1987. Google Scholar [9] R.V. Chacon and N.A. Friedman, Approximation and invariant measures, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 3 (1965), 286-295. doi: 10.1007/BF00535780. Google Scholar [10] T. Chow, A spectral theory for direct integrals of operators, Math. Ann., 188 (1970), pp. 285-303. doi: 10.1007/BF01431463. Google Scholar [11] M. Cowling and T. Steger, The irreducibility of restrictions of unitary representations to lattices, J. Reine Angew. Math., 420 (1991), 85-98. Google Scholar [12] J. Dixmier, Les C*-algèbres et leurs représentations, Gauthier-Villars 1969. Google Scholar [13] A. Dudko, R. Grigorchuk, On diagonal actions of branch groups and the corresponding characters, arxiv: 1412.5476.Google Scholar [14] A. Dudko, R. Grigorchuk, On irreducibility and disjointness of Koopman and quasi-regular representations of weakly branch groups, Modern Theory of Dynamical Systems: A Tribute to Dmitry Victorovich Anosov, AMS Cont. Math. Ser. (2016), to appear.Google Scholar [15] A. Dudko and K. Medynets, On Characters of Inductive Limits of Symmetric Groups, Journal of Functional Analysis,, 264 (2013), 1565-1598. doi: 10.1016/j.jfa.2013.01.013. Google Scholar [16] J. Feldman and C. Moore, Ergodic equivalence relations. cohomology, and von Neumann algebras.Ⅰ, Trans. Amer. Math. Soc., 234 (1977), 289-324. doi: 10.1090/S0002-9947-1977-0578656-4. Google Scholar [17] J. Feldman and C. Moore, Ergodic equivalence relations. cohomology, and von Neumann algebras. Ⅱ, Trans. Amer. Math. Soc., 234 (1977), 325-359. doi: 10.1090/S0002-9947-1977-0578730-2. Google Scholar [18] A. Figá-Talamanca and M. A. Picardello, Harmonic analysis on free groups, Lecture Notes in Pure and Applied Mathematics, vol. 87, Marcel Dekker Inc., New York, (1983). Google Scholar [19] A. Figá-Talamanca, T. Steger, Harmonic analysis for anisotropic random walks on homogeneous trees, Mem. Amer. Math. Soc. 110 (1994), no. 531, ⅶ+68. Google Scholar [20] E. Glasner, Ergodic theory via joinings, Math. Surv. and Mon., 101, Amer. Math. Soc. (2003). Google Scholar [21] R. I. Grigorchuk, Burnside's Problem on Periodic Groups, Funkts. Anal. Prilozh. 14 (1), 53-54 (1980)[Funct. Anal. Appl. 14, 41-43 (1980)]. Google Scholar [22] R. I. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 48 (1984), 939-985. Google Scholar [23] R. I. Grigorchuk, Degrees of growth of p-groups and torsion-free groups (Russian) Mat. Sb. (N. S. ) 126 (168) (1985),194-214. Google Scholar [24] R. I. Grigorchuk, Just Infinite Branch Groups, New Horizons in Pro-p Groups (Birkhauser, Boston, MA, 2000), Prog. Math. 184, pp. 121-179. Google Scholar [25] R. I. Grigorchuk, Some topics in the dynamics of group actions on rooted trees, Proceedings of the Steklov Institute of Mathematics, 273, Issue 1 (2011), pp. 64-175. Google Scholar [26] R. I. Grigorchuk, D. Lenz and T. Smirnova-Nagnibeda, Spectra of Schreier graphs of Grigorchuk's group and Schroedinger operators with aperiodic order, arXiv: 1412.6822.Google Scholar [27] R. I. Grigorchuk and V. Nekrashevich, Amenable actions of non-amenable groups, J. of Math. Sci., 140 (2007), pp. 391-397. Google Scholar [28] R. Grigorchuk, V. Nekrashevich, and Z. Sunic, From self-similar groups to self-similar sets and spectra, Fractal Geometry and Stochastics V, 70, series Progress in Probability, pp. 175-207. doi: 10.1007/978-3-319-18660-3_11. Google Scholar [29] R. I. Grigorchuk, V. V. Nekrashevich, and V. I. Sushchanskii, Automata, Dynamical Systems, and Groups, Tr. Mat. Inst. im. V. A. Steklova, Ross. Akad. Nauk 231,134-214 (2000)[Proc. Steklov Inst. Math. 231,128-203 (2000)]. Google Scholar [30] R. I. Grigorchuk and Ya. Krylyuk, The spectral measure of the Markov operator related to 3-generated 2-group of intermediate growth and its Jacobi parameters, Algebra Discrete Math., 13 (2012), pp. 237-272. Google Scholar [31] P. de la Harpe, On simplicity of reduced C*-Ualgebras of groups, Bull. London Math. Soc., 39 (2007), pp. 1-26. doi: 10.1112/blms/bdl014. Google Scholar [32] N. Higson and G. Kasparov, Operator K-theory for groups which act properly and isometrically on Hilbert space, Electron. Res. Announc. Amer. Math. Soc., 3 (1997), 131-142. doi: 10.1090/S1079-6762-97-00038-3. Google Scholar [33] R. Kadison and J. Ringrose, Fundamentals of the theory of operator algebras. Vol. Ⅰ. Elementary theory. Pure and Applied Mathematics, 100. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. Google Scholar [34] R. Kadison and J. Ringrose, Fundamentals of the theory of operator algebras. Vol. Ⅱ. Advanced theory. Pure and Applied Mathematics, 100. Academic Press, Inc., Orlando, FL, 1986. Google Scholar [35] I. Lysenok, A set of defining relations for the Grigorchuk group, Mat. Zametki, 38 (1985), pp. 503U516.. Google Scholar [36] S. Kerov and A. Vershik, Characters and factor representations of the infinite symmetric group, Soviet Math. Dokl., 23 (1981), 389-392. Google Scholar [37] H. Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc., 92 (1959), 336-354. doi: 10.1090/S0002-9947-1959-0109367-6. Google Scholar [38] G. Kuhn, Amenable actions and weak containment of certain representations of discrete groups, Proc. Amer. Math. Soc., 122 (1994), 751-757. doi: 10.1090/S0002-9939-1994-1209424-3. Google Scholar [39] G. Kuhn and T. Steger, More irreducible boundary representations of free groups, Duke Math. J., 82, no. 2 (1996), 381-435. doi: 10.1215/S0012-7094-96-08218-6. Google Scholar [40] G. W. Mackey, The Theory of Unitary Group Representations, Univ. Chicago Press, Chicago, 1976, Chicago Lect. Math. Google Scholar [41] M. Naimark, Normed rings, Noordhoff, Groningen, 1959, xvi+560pp. Google Scholar [42] V. Nekrashevych, Self-similar Groups, Am. Math. Soc., Providence, RI, 2005, Math. Surv. Monogr. 117. Google Scholar [43] M. Pichot, Sur la théorie spectrale des relations d'équivalence mesurées, Journal of the Inst. of Math. Jussieu, 3, 6 (2007), 453-500. doi: 10.1017/S1474748007000096. Google Scholar [44] D. Powers, Simplicity of the C*-algebra associated with the free group on two generators, Duke Math. J., 42 (1975), pp. 151-156. doi: 10.1215/S0012-7094-75-04213-1. Google Scholar [45] V. A. Rokhlin, On the basic ideas of measure theory, Mat. Sb. 25 (1) (1949), 107-150. Engl. transl., On the fundamental ideas of measure theory, (Am. Math. Soc., Providence, RI, 1952), AMS Transl., No. 71, 55pp. Google Scholar [46] V. A. Rokhlin, Selected topics from the metric theory of dynamical systems, (Russian) Us-pekhi Mat. Nauk, 4 (1949), 57–128 Google Scholar [47] H. L. Skudlarek, Die unzerlegbaren Charaktere einiger diskreter Gruppen, Math. Ann., 223 (1976), 213-231. doi: 10.1007/BF01360954. Google Scholar [48] M. Takesaki, Theory of Operator Algebras. I, Encyclopedia of Mathematical Sciences, 124, Operator Algebras and Non-commutative Geometry, 5, Springer-Verlag, Berlin, 2002. Google Scholar [49] M. Takesaki, Theory of Operator Algebras. III, Encyclopedia of Mathematical Sciences, 127, Operator Algebras and Non-commutative Geometry, 8, Springer-Verlag, Berlin, 2003. Google Scholar [50] R.J. Zimmer, Hyperfinite factors and amenable ergodic actions, Inv. Math., 41 (1977), 23-31. doi: 10.1007/BF01390162. Google Scholar
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