Recent Results and Open Questions Relating Chu Duality and Bohr Compactifications of Locally Compact Groups

Open Problems in Topology II

Volumn , Issue , 2007, Pages 407-422

  Galindo, Jorge ^a   Hernández, Salvador ^a   Wu, Ta Sun ^b  

^a UNIVERSITAT JAUME I   (Spain)

^b CASE WESTERN RESERVE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84882839172     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1016/B978-044452208-5/50042-9     Document Type: Chapter

References (62)

1. Baggett L. A separable group having a discrete dual space is compact. J. Punct. Anal. 1972, 10:131-148.
2. Bourgain J. Propriétés de décomposition pour les ensembles de Sidon. Bull. Soc. Math. France 1983, 111(no. 4):421-428.
3. 1 sequences generated by Sidon sets. J. London Math. Soc. 1984, 29(no. 2):283-288.
4. Chou C. Weakly almost periodic functions and thin sets in discrete groups. Trans. Amer. Math. Soc. 1990, 321(no. 1):333-346.
5. Chu H. Gompactification and duality of topological groups. Trans. Amer. Math. Soc. 1966, 123:310-324.
6. Comfort W.W., Hernández S., Trigos-Arrieta F.J. Relating a locally compact abelian group to its Bohr compactification. Adv. Math. 1996, 120(no. 2):322-344.
7. Comfort W.W., Hernández S., Trigos-Arrieta F.J. Gross sections and homeomorphism classes of abelian groups equipped with the Bohr topology. Topology Appl. 2001, 115(no. 2):215-233.
8. Comfort W.W., Remus D., Szambien H. Extending ring topologies. J. Algebra 2000, 232(no. 1):21-47.
9. De-Michele L., Soardi P.M. Existence of Sidon sets in discrete FC-groups. Proc. Amer. Math. Soc. 1976, 55(no. 2):457-460.
10. Dikranjan D., Watson S. A solution to van Douwen's problem on Bohr topologies. J. Pure Appl. Algebra 2001, 163(no. 2):147-158.
11. van Douwen E.K. The maximal totally bounded group topology on G, the biggest minimal G-space for abelian groups G. Topology Appl. 1990, 34(no. 1):69-91.
12. Dunkl C.F., Ramirez D.E. Topics in harmonic analysis 1971, Appleton-Century-Croft, New York.
13. Enock M., Schwartz J.-M. Kac algebras and duality of locally compact groups 1992, Springer, Berlin.
14. Eymard P. L'algèbre de Fourier d'un groupe localement compact. Bull. Soc. Math. France 1964, 92:181-236.
15. Ferri S., Galindo J. On the WAP-compactification of a topological group 2006, Preprint.
16. Galindo J., Hernández S. On a theorem of van Douwen. Extraeta Math. 1998, 13(no. 1):115-123.
17. Galindo J., Hernández S. The concept of boundedness and the Bohr compactification of a MAP abelian group. Fund. Math. 1999, 159(no. 3):195-218.
18. Galindo J., Hernández S. Interpolation sets and the Bohr topology of locally compact groups. Adv. Math. 2004, 188(no. 1):51-68.
19. Glicksberg I. Uniform boundedness for groups. Canad. J. Math. 1962, 14:269-276.
20. Goto M. Note on a topology of a dual space. Proc. Amer. Math. Soc. 1961, 12:41-46.
21. 0 sets in abelian groups. III. Interpolation by measures on small sets. Studia Math. 2005, 171(no. 1):15-32.
22. 0-sets. Colloq. Math. 1987, 53(no. 2):269-270.
23. 0 sets in non-abelian groups. Math. Proc. Cambridge Philos. Soc. 2003, 135(no. 1):81-98.
24. de la Harpe P., Valette A. La propriété (T) de Kazhdan pour les groupes localement compacts. Astérisque 1989, (no. 175). 158 pp.
25. Hart J.E., Kunen K. Bohr compactifications of discrete structures. Fund. Math. 1999, 160(no. 2):101-151.
26. Hart J.E., Kunen K. Bohr compactifications of non-abelian groups. Topology Proc. 2001, 26(no. 2):593-626. 02.
27. Hart J.E., Kunen K. Bohr topologies and compact function spaces. Topology Appl. 2002, 125(no. 2):183-198.
28. Hartman S., Ryll-Nardzewski C. Almost periodic extensions of functions. Colloq. Math. 1964, 12:23-39.
29. Hartman S., Ryll-Nardzewski C. Almost periodic extensions of functions. II. Colloq. Math. 1966, 15:79-86.
30. Hernández S. The dimension of an LCA group in its Bohr topology. Topology Appl. 1998, 86(no. 1):63-67.
31. Hernández S., Wu T.S. The Bohr topology of non abelian groups 2005, Preprint.
32. Hernández S., Wu T.S. Some new results on the Chu duality of discrete groups. Monatsh. Math. 2006, Article in press.
33. Hewitt E., Ross K.A. Abstract harmonic analysis 1970, Vol. II. Springer, Berlin.
34. Heyer H. Dualität lokalkompakter Gruppen. Lecture Notes in Math. 1970, vol. 150. Springer, Berlin.
35. Heyer H. Groups with Chu duality, Probability and information theory, II. Lecture Notes in Math. 1973, vol. 296:181-215. Springer, Berlin.
36. Katznelson Y. Sequences of integers dense in the Bohr group. Proc. Roy. Inst. of Tech. 1973, (no. June):79-86.
37. Kunen K. Bohr topologies and partition theorems for vector spaces. Topology Appl. 1998, 90:97-107.
38. Kunen K., Rudin W. Lacunarity and the Bohr topology. Math. Proc. Cambridge Philos. Soc. 1999, 126(no. 1):117-137.
39. Landstad M.B. On the Bohr topology in amenable topological groups. Math. Scand. 1971, 28:207-214.
40. Lefèvre P., Rodríguez-Piazza L. The union of a Riesz set and a Lust-Piquard set is a Riesz set. J. Funct. Anal. 2006, 233(no. 2):545-560.
41. Lopez J.M., Ross K.A. Sidon sets 1975, Marcel Dekker Inc., New York.
42. Mayer M. Strongly mixing groups. Semigroup Forum 1997, 54(no. 3):303-316.
43. Moran W. On almost periodic compactifications of locally compact groups. J. London Math. Soc. 1971, 3:507-512.
44. Paterson A.L.T. Amenability 1988, American Mathematical Society, Providence, RI.
45. Picardello M.A. Lacunary sets in discrete noncommutative groups. Univ. Genova Pubbl. 1st. Mat. 1973, 2(no. 60):ii+24.
46. Poguntke D. A universal property of the Takahashi quasi-dual. Canad. J. Math. 1972, 24:530-536.
47. Poguntke D. Chu-Dualität und zwei Klassen maximal fastperiodischer Gruppen. Monatsh. Math. 1976, 82(no. 1):31-50.
48. Poguntke D. Zwei Klassen lokalkompakter maximal fastperiodischer Gruppen. Monatsh. Math. 1976, 81(no. 1):15-40.
49. Ramsey L.T. Bohr cluster points of Sidon sets. Colloq. Math. 1995, 68(no. 2):285-290.
50. 0 sets. Colloq. Math. 1996, 70(no. 1):103-132.
51. Remus D., Trigos-Arrieta F.J. The Bohr topology of Moore groups. Topology Appl. 1999, 97(no. 1-2):85-98.
52. Riggins L. On infinite groups and unitary duality. Ph. D. thesis 1998, Case Western Reserve University.
53. Roeder D.W. A characterization of unitary duality. Trans. Amer. Math. Soc. 1970, 148:129-135.
54. 1. Proc. Natl. Acad. Sci. 1974, 71:2411-2413.
55. Rudin W. Fourier analysis on groups 1990, John Wiley & Sons Inc., New York.
56. Ruppert W.A.F. On weakly almost periodic sets. Semigroup Forum 1985, 32(no. 3):2411-2413.
57. Strzelecki E. On a problem of interpolation by periodic and almost periodic functions. Col-loq. Math. 1963, 11:91-99.
58. Takahashi S. A duality theorem for representable locally compact groups with compact com-mutator subgroup. Tôhoku Math. J. 1952, 4(2):115-121.
59. Tits J. Free subgroups in linear groups. J. Algebra 1972, 20:250-270.
60. van der Waerden B.L. Stetigkeitssätze für halbeinfache Liesche Gruppen. Math. Z. 1933, 36(no. 1):780-786.
61. Wang P.S. On isolated points in the dual spaces of locally compact groups. Math. Ann. 1975, 218(no. 1):19-34.
62. Wu T.S., Riggins L. Maximally almost periodic groups and a theorem of Glicksberg. Ann. New York Acad. Sei. 1996, vol. 806:454-464. New York Acad. Sci.