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Representations of dynamical systems on Banach spaces not containing $ l_1$


Authors: E. Glasner and M. Megrelishvili
Journal: Trans. Amer. Math. Soc. 364 (2012), 6395-6424
MSC (2010): Primary 37Bxx, 54H20, 54H15, 46-xx
DOI: https://doi.org/10.1090/S0002-9947-2012-05549-8
Published electronically: July 11, 2012
MathSciNet review: 2958941
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Abstract: For a topological group $ G$, we show that a compact metric $ G$-space is tame if and only if it can be linearly represented on a separable Banach space which does not contain an isomorphic copy of $ l_1$ (we call such Banach spaces, Rosenthal spaces). With this goal in mind we study tame dynamical systems and their representations on Banach spaces.


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  • 1. E. Akin, J. Auslander, and K. Berg, Almost equicontinuity and the enveloping semigroup, Topological dynamics and applications, Contemporary Mathematics 215, a volume in honor of R. Ellis, 1998, pp. 75-81. MR 1603149 (99k:54032)
  • 2. J. Bourgain, D.H. Fremlin and M. Talagrand, Pointwise compact sets in Baire-measurable functions, Amer. J. of Math., 100:4 (1977), 845-886. MR 509077 (80b:54017)
  • 3. R.D. Bourgin, Geometric Aspects of Convex Sets with the Radon-Nikodým Property, Lecture Notes in Math. 993, Springer-Verlag, 1983. MR 704815 (85d:46023)
  • 4. W.J. Davis, T. Figiel, W.B. Johnson and A. Pelczyński, Factoring weakly compact operators, J. of Funct. Anal., 17 (1974), 311-327. MR 0355536 (50:8010)
  • 5. D. van Dulst, Characterization of Banach spaces not containing $ l^1$, Centrum voor Wiskunde en Informatica, Amsterdam, 1989. MR 1002733 (90h:46037)
  • 6. R. Ellis, Lectures on Topological Dynamics, W. A. Benjamin, Inc., New York, 1969. MR 0267561 (42:2463)
  • 7. R. Ellis and M. Nerurkar, Weakly almost periodic flows, Trans. Amer. Math. Soc. 313, (1989), 103-119. MR 930084 (89i:28010)
  • 8. R. Engelking, General topology, Heldermann Verlag, Berlin, 1989. MR 1039321 (91c:54001)
  • 9. M. Fabian, Gateaux differentiability of convex functions and topology. Weak Asplund spaces, Canadian Math. Soc. Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, New York, 1997. MR 1461271 (98h:46009)
  • 10. E. Glasner, On tame dynamical systems, Colloq. Math. 105 (2006), 283-295. MR 2237913 (2007d:37005)
  • 11. E. Glasner, The structure of tame minimal dynamical systems, Ergod. Th. and Dynam. Sys. 27, (2007), 1819-1837. MR 2371597 (2008m:37015)
  • 12. E. Glasner, Enveloping semigroups in topological dynamics, Topology Appl. 154, (2007), 2344-2363. MR 2328017 (2008f:37021)
  • 13. E. Glasner and M. Megrelishvili, Hereditarily non-sensitive dynamical systems and linear representations, Colloq. Math., 104 (2006), no. 2, 223-283. MR 2197078 (2006m:37009)
  • 14. E. Glasner and M. Megrelishvili, New algebras of functions on topological groups arising from $ G$-spaces, Fundamenta Math., 201 (2008), 1-51. MR 2439022 (2010f:37015)
  • 15. E. Glasner and M. Megrelishvili, On fixed point theorems and nonsensitivity, Israel J. of Math. (to appear).
  • 16. E. Glasner, M. Megrelishvili and V.V. Uspenskij, On metrizable enveloping semigroups, Israel J. of Math. 164 (2008), 317-332. MR 2391152 (2009a:37013)
  • 17. E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity 6, (1993), 1067-1075. MR 1251259 (94j:58109)
  • 18. G. Godefroy, Compacts de Rosenthal, Pacific J. Math., 91 (1980), 293-306. MR 615679 (82f:54030)
  • 19. M.M. Guillermo, Indice de K-determinación de espacios topológicos y $ \sigma $-fragmentabilidad de aplicaciones, Tesis Doctoral, Universidad de Murcia, Spain, 2003.
  • 20. W. Huang, Tame systems and scrambled pairs under an abelian group action, Ergod. Th. Dynam. Sys. 26 (2006), 1549-1567. MR 2266373 (2007j:37012)
  • 21. R.C. James, A separable somewhat reflexive Banach space with nonseparable dual, Bull. Amer. Math. Soc., 80 (1974), 738-743. MR 0417763 (54:5811)
  • 22. J.E. Jayne, J. Orihuela, A.J. Pallares and G. Vera, $ \sigma $-fragmentability of multivalued maps and selection theorems, J. Funct. Anal. 117 (1993), no. 2, 243-273. MR 1244937 (94m:46023)
  • 23. J.E. Jayne and C.A. Rogers, Borel selectors for upper semicontinuous set-valued maps, Acta Math. 155 (1985), 41-79. MR 793237 (87a:28011)
  • 24. A.S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, 156, Springer-Verlag, 1991. MR 1321597 (96e:03057)
  • 25. D. Kerr and H. Li, Independence in topological and $ C^*$-dynamics, Math. Ann. 338 (2007), 869-926. MR 2317754 (2009a:46126)
  • 26. A. Köhler, Enveloping semigrops for flows, Proceedings of the Royal Irish Academy, 95A (1995), 179-191. MR 1660377 (99i:47056)
  • 27. J. Lindenstrauss and C. Stegall, Examples of separable spaces which do not contain $ l_1$ and whose duals are nonseparable, Studia Math., 54 (1975), 81-105. MR 0390720 (52:11543)
  • 28. M. Megrelishvili, Fragmentability and continuity of semigroup actions, Semigroup Forum, 57 (1998), 101-126. MR 1621881 (99c:54053)
  • 29. M. Megrelishvili, Operator topologies and reflexive representability, In: ``Nuclear groups and Lie groups'' Research and Exposition in Math. series, vol. 24, Heldermann Verlag, Berlin, 2001, 197-208. MR 1858149 (2002m:46027)
  • 30. M. Megrelishvili, Fragmentability and representations of flows, Topology Proceedings, 27:2 (2003), 497-544. See also: www.math.biu.ac.il/ $ {\tilde {}}$ megereli. MR 2077804 (2005h:37042)
  • 31. E. Michael and I. Namioka, Barely continuous functions, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys., 24 (1976), 889-892. MR 0431092 (55:4094)
  • 32. I. Namioka, Separate continuity and joint continuity, Pacific. J. Math., 51 (1974), 515-531. MR 0370466 (51:6693)
  • 33. I. Namioka, Radon-Nikodým compact spaces and fragmentability, Mathematika 34, (1987), 258-281. MR 933504 (89i:46021)
  • 34. I. Namioka and R.R. Phelps, Banach spaces which are Asplund spaces, Duke Math. J., 42 (1975), 735-750. MR 0390721 (52:11544)
  • 35. E. Odell and H. P. Rosenthal, A double-dual characterization of separable Banach spaces containing $ l\sp {1}$, Israel J. Math., 20 (1975), 375-384. MR 0377482 (51:13654)
  • 36. R. Pol, Note on Compact Sets of First Baire Class Functions, Proc. Amer. Math. Soc. 96, No. 1. (1986), pp. 152-154. MR 813828 (87b:54033)
  • 37. L.H. Riddle, E. Saab and J.J. Uhl, Sets with the weak Radon-Nikodým property in dual Banach spaces, Indiana Univ. Math. J., 32 (1983), 527-541. MR 703283 (84h:46028)
  • 38. H.P. Rosenthal, A characterization of Banach spaces containing $ l_1$, Proc. Nat. Acad. Sci. U.S.A., 71 (1974), 2411-2413. MR 0358307 (50:10773)
  • 39. H.P. Rosenthal, Point-wise compact subsets of the first Baire class, Amer. J. of Math., 99:2 (1977), 362-378. MR 0438113 (55:11032)
  • 40. H.P. Rosenthal, Some recent discoveries in the isomorphic theory of Banach spaces, Bull. Amer. Math. Soc. 84:5 (1978), 803-831. MR 499730 (80d:46023)
  • 41. E. Saab and P. Saab, A dual geometric characterization of Banach spaces not containing $ l_1$, Pacific J. Math., 105:2 (1983), 413-425. MR 691612 (85c:46013)
  • 42. M. Talagrand, Pettis integral and measure theory, Mem. Amer. Math. Soc. No. 51 (1984). MR 756174 (86j:46042)
  • 43. S. Todorcević, Topics in topology, Lecture Notes in Mathematics, 1652, Springer-Verlag, 1997. MR 1442262 (98g:54002)

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Additional Information

E. Glasner
Affiliation: Department of Mathematics, Tel-Aviv University, Tel Aviv, Israel
Email: glasner@math.tau.ac.il

M. Megrelishvili
Affiliation: Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel
Email: megereli@math.biu.ac.il

DOI: https://doi.org/10.1090/S0002-9947-2012-05549-8
Keywords: Baire one function, Banach representation of dynamical systems, enveloping semigroup, fragmentability, Rosenthal’s dichotomy, Rosenthal’s compact, Tame system
Received by editor(s): April 16, 2008
Received by editor(s) in revised form: November 9, 2009, and January 21, 2011
Published electronically: July 11, 2012
Additional Notes: The first author’s research was partially supported by BSF (Binational USA-Israel) grant no. 2006119.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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