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A decomposition theorem for unitary group representations on Kaplansky–Hilbert modules and the Furstenberg–Zimmer structure theorem

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Abstract

In this paper, a decomposition theorem for (covariant) unitary group representations on Kaplansky–Hilbert modules over Stone algebras is established, which generalizes the well-known Hilbert space case (where it coincides with the decomposition of Jacobs, deLeeuw and Glicksberg).

The proof rests heavily on the operator theory on Kaplansky–Hilbert modules, in particular the spectral theorem for Hilbert–Schmidt homomorphisms on such modules.

As an application, a generalization of the celebrated Furstenberg–Zimmer structure theorem to the case of measure-preserving actions of arbitrary groups on arbitrary probability spaces is established.

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References

  1. A. Bátkai, U. Groh, D. Kunszenti-Kovács, and M. Schreiber, Decomposition of operator semigroups on \({\rm W}^*\)-algebras, Semigroup Forum, 84 (2012), 8-24.

  2. P. Cheridito, M. Kupper, and N. Vogelpoth, Conditional analysis on \(\mathbb R^d\), in: Set Optimization and Applications-the State of the Art, Springer Proceedings in Mathematics and Statistics, vol. 151, Springer (2015), pp. 179-211.

  3. J. B. Conway, A Course in Functional Analysis, Graduate Texts in Mathematics, vol. 96, Springer (1985).

  4. J. Dixmier and A. Douady, Champs continus dèspaces hilbertiens et de \(C^{\ast}\)-alg`ebres, Bull. Soc. Math. France, 91 (1963), 227-284.

  5. H. G. Dales, F. K. Dashiell, Jr., A. T.-M. Lau, and D. Strauss, Banach Spaces of Continuous Functions as Dual Spaces, CMS Books in Mathematics, Springer (2016).

  6. M. J. Duprè and R. M. Gillette, Banach Bundles, Banach Modules and Automorphisms of C*-Algebras, Pitman (1983).

  7. J. Dixmier, C*-Algebras, North-Holland (1977).

  8. S. Drapeau, A. Jamneshan, M. Karliczek, and M. Kupper, The algebra of conditional sets and the concepts of conditional topology and compactness, J. Math. Anal. Appl., 437 (2016), 561-589.

  9. R. Derndinger, R. Nagel, and G. Palm, Ergodic Theory in the Perspective of Functional Analysis, manuscript (1987).

  10. D. Deckard and C. Pearcy, On matrices over the ring of continuous complex valued functions on a Stonian space, Proc. Amer. Math. Soc., 14 (1963), 322-328.

  11. D. Deckard and C. Pearcy, On continuous matrix-valued functions on a Stonian space, Pacific J. Math., 14 (1964), 857-869.

  12. N. Edeko, The Furstenberg-Zimmer structure theorem for stationary random walks, arXiv:2212.04353 (2022).

  13. N. Edeko, A. Jamneshan, and H. Kreidler, A Peter-Weyl theorem for compact group bundles and the geometric representation of relatively ergodic compact extensions, arXiv:2302.13630 (2023).

  14. N. Edeko and H. Kreidler, Distal systems in topological dynamics and ergodic theory, Ergodic Theory Dynam. Systems, 43 (2023), 2651-2672.

  15. T. Eisner, A view on multiple recurrence, Indag. Math. (N. S.), 34 (2023), 231-247.

  16. T. Eisner, B. Farkas, M. Haase, and R. Nagel, Operator Theoretic Aspects of Ergodic Theory, Graduate Texts in Mathematics, vol. 272, Springer (2015).

  17. R. Ellis, Topological dynamics and ergodic theory, Ergodic Theory Dynam. Systems, 7 (1987), 25-47.

  18. M. Einsiedler and T. Ward, Ergodic Theory, Graduate Texts in Math., vol. 259, Springer (2011).

  19. M. Frank, Hilbert C*-modules over monotone complete C*-algebras, Math. Nachr., 175 (1995), 61-83.

  20. H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for commutingtransformations, J. Anal. Math., 34 (1978), 275-291.

  21. H. Furstenberg, Y. Katznelson, and D. Ornstein, The ergodic theoretical proof of Szemerédi's theorem, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 527-552.

  22. D. Filipović, M. Kupper, and N. Vogelpoth, Separation and duality in locally \(L^0\)- convex modules, J. Funct. Anal., 256 (2009), 3996-4029.

  23. H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Anal. Math., 31 (1977), 204-256.

  24. H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press (1981).

  25. G. Gierz, Bundles of Topological Vector Spaces and Their Duality, Lecture Notes in Math., vol. 955, Springer (1982).

  26. L. Gillman and M. Jerison, Rings of Continuous Functions, University Series in Higher Math., Springer (1976).

  27. E. Glasner, Ergodic Theory via Joinings, Mathematical Surveys and Monographs, vol. 101, American Mathematical Society (2003).

  28. R. Godement, Sur la théorie des représentations unitaires, Ann. of Math. (2), 53(1951), 68-124.

  29. U. Gönüllü, Schatten-type classes of operators on Kaplansky-Hilbert modules, www.researchgate.net/publication/311650264 (2014).

  30. U. Gönüllü, A representation of cyclically compact operators on Kaplansky-Hilbert modules, Arch. Math. (Basel), 106 (2016), 41-51.

  31. A. E. Gutman, Banach bundles in the theory of lattice normed spaces. I. Continuous Banach bundles, Siberian Adv. Math., 3 (1993), 1-55.

  32. G. L. M. Groenewegen and A. C. M. van Rooij, Spaces of Continuous Functions, Atlantis Studies in Math., vol. 4, Atlantis Press (2016).

  33. M. Haase and H. Kreidler, Tensor products of Kaplansky-Hilbert modules and discrete spectrum in conditional products (in preparation).

  34. K. H. Hofmann and K. Keimel, Sheaf theoretical concepts in analysis: Bundles and sheaves of Banach spaces, Banach \(\mathrm{C}({X})\)-modules, in: Applications of Sheaves, Lecture Notes in Math., vol. 753, Springer (1977), pp. 415-441.

  35. A. Jamneshan, An uncountable Furstenberg-Zimmer structure theory, Ergodic Theory Dynam. Systems, 43 (2023), 2404-2436.

  36. A. Jamneshan and P. Spaas, On compact extensions of tracial W*-dynamical systems, arXiv:2212.04353 (2022).

  37. A. Jamneshan and T. Tao, An uncountable Mackey-Zimmer theorem, Studia Math., 266 (2022), 241-289.

  38. A. Jamneshan and T. Tao, An uncountable Moore-Schmidt theorem, Ergodic Theory Dynam. Systems, 43 (2023), 2376-2403.

  39. A. Jamneshan and T. Tao, Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration, 2020, Fund. Math., 261 (2023), 1-98.

  40. A. Jamneshan and J. M. Zapata, On compactness in \(L^0\)-modules, arXiv:1711.09785 (2017).

  41. S. Kakutani, Concrete representation of abstract \((L)\)-spaces and the mean ergodic theorem, Ann. of Math. (2), 42 (1941), 523-537.

  42. S. Kakutani, Concrete representation of abstract \((M)\)-spaces. (A characterization of the space of continuous functions.), Ann. of Math. (2), 42 (1941), 994-1024.

  43. I. Kaplansky, Projections in Banach algebras, Ann. of Math. (2), 53 (1951), 235-249.

  44. I. Kaplansky, Modules over operator algebras, Amer. J. Math., 75 (1953), 839-858.

  45. A. G. Kusraev and S. S. Kutateladze, Boolean Valued Analysis, Kluwer Academic Publishers (1999).

  46. D. Kerr and H. Li, Ergodic Theory, Springer Monographs in Math., Springer (2016).

  47. H. Kreidler and S. Siewert, Gelfand-type theorems for dynamical Banach modules, Math. Z., 297 (2021), 1353--1382.

  48. A. G. Kusraev, Cyclically compact operators in Banach-Kantorovich spaces, in: Partial Differential Equations (Novosibirsk, 1983), "Nauka" Sibirsk. Otdel. (1986), pp. 108-116,221.

  49. A. G. Kusraev, Cyclically compact operators in Banach spaces, Vladikavkaz. Mat. Zh., 2 (2000), 10-23.

  50. A. G. Kusraev, Dominated Operators, Mathematics and Its Applications, vol. 519, Springer (2000).

  51. C. E. Lance, Hilbert C*-Modules, Cambridge University Press (1995).

  52. M. Lemańczyk, J.-P. Thouvenot, and B. Weiss, Relative discrete spectrum and joinings, Monatsh. Math., 137 (2002), 57-75.

  53. J. Moreira, F. K. Richter, and D. Robertson, A proof of a sumset conjecture of Erdős, Ann. of Math. (2), 189 (2019), 605-652.

  54. M. Ozawa, Boolean valued interpretation of Hilbert space theory, J. Math. Soc. Japan, 35 (1983), 609-627.

  55. M. Ozawa, Boolean valued interpretation of Banach space theory and module structures of von Neumann algebras, Nagoya Math. J., 117 (1990), 1-36.

  56. G. K. Pedersen, \(C^{\ast} \)-Algebras and Their Automorphism Groups, Academic Press (1979).

  57. H. Schaefer, Banach Lattices and Positive Operators, Grundlehren der mathematischen Wissenschaften, vol. 215, Springer (1974).

  58. M. Schreiber, Uniform families of ergodic operator nets, Semigroup Forum, 86 (2013), 321-336.

  59. M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc., 41 (1937), 375-481.

  60. R. G. Swan, Vector bundles and projective modules, Trans. Amer. Math. Soc., 105 (1962), 264-277.

  61. H. Takemoto, On a characterization of AW\(^{\ast} \)-modules and a representation of Gelfand type of noncommutative operator algebras, Michigan Math. J., 20 (1973), 115- 127.

  62. G. Takeuti, Two Applications of Logic to Mathematics, Princeton University Press (1978).

  63. G. Takeuti, Boolean valued analysis, in: Applications of Sheaves, Lecture Notes in Math., vol. 753, Springer (1979), pp. 714-731.

  64. G. Takeuti, A transfer principle in harmonic analysis, J. Symbolic Logic, 44 (1979), 417-440.

  65. M. Takesaki, Theory of Operator Algebras. I, Springer (2002).

  66. T. Tao, Poincaré's Legacies, Part I, American Mathematical Society (2009).

  67. J. D. M. Wright, Stone-algebra-valued measures and integrals, Proc. London Math. Soc. (3), 19 (1969), 107-122.

  68. J. D. M. Wright, A spectral theorem for normal operators on a Kaplansky-Hilbert module, Proc. London Math. Soc. (3), 19 (1969), 258-268.

  69. R. J. Zimmer, Extensions of ergodic group actions, Illinois J. Math., 20 (1976), 373-409.

  70. R. J. Zimmer, Ergodic actions with generalized discrete spectrum, Illinois J. Math., 20 (1976), 555-588.

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Acknowledgements

We thank Rainer Nagel for long-lasting support and many inspiring discussions. We thank Terence Tao and Asgar Jamneshan for becoming interested in our work and for their readiness to exchange ideas on these topics, and Patrick Hermle and Sascha Trostorff for pointing out mistakes in an earlier version of this article. We are also grateful towards the anonymous referees for their valuable feedback.

Nikolai Edeko and Henrik Kreidler are grateful to the MFO for providing an opportunity to work on this project in a stimulating atmosphere.

Parts of this work were conceived when Markus Haase spent a research semester at UNSW, Sydney. He gratefully acknowledges the kind invitation by Fedor Sukochev, inspiring discussions with Thomas Scheckter and he also thanks Uğur Gönüllü for some helpful conversations about cyclical compactness.

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Authors and Affiliations

  1. Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

    N. Edeko

  2. Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Heinrich-Hecht-Platz 6, D-24118 Kiel, Germany

    M. Haase

  3. Fachgruppe Mathematik und Informatik, Bergische Universität Wuppertal, Gaußstraße 20, D-42119 Wuppertal, Germany

    H. Kreidler

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  1. N. Edeko
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Correspondence to H. Kreidler.

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To Rainer Nagel in tribute to his tremendous achievements within and outside mathematics.

Markus Haase acknowledges the financial support by the DFG (project number 431663331).

Henrik Kreidler acknowledges the financial support by the DFG (project number 451698284).

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Edeko, N., Haase, M. & Kreidler, H. A decomposition theorem for unitary group representations on Kaplansky–Hilbert modules and the Furstenberg–Zimmer structure theorem. Anal Math (2024). https://doi.org/10.1007/s10476-024-00020-1

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