Abstract.
This paper deals mainly with the Chu duality of discrete groups. Among other results, we give sufficient conditions for an FC group to satisfy Chu duality and characterize when the Chu quasi-dual and the Takahashi quasi-dual of a group G coincide. As a consequence, it follows that when G is a weak sum of a family of finite simple groups, if the exponent of the groups in the family is bounded then G satisfies Chu duality; on the other hand, if the exponent of the group goes to infinity, then the Chu quasi-dual of G coincides with its Takahashi quasi-dual. We also present examples of discrete groups whose Chu quasi-duals are not locally compact and examples of discrete Chu reflexive groups which contain non-trivial sequences converging in the Bohr topology of the groups. Our results systematize some previous work and answer some open questions on the subject [2, 16, 3].
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References
N Bourbaki (1971) Topologie générale, Chap 1–4 Hermann Paris
H Chu (1966) ArticleTitleCompactification and duality of topological groups Trans Amer Math Soc 123 310–324 Occurrence Handle0158.03002 Occurrence Handle195988 Occurrence Handle10.2307/1994659
Comfort WW, Salvador Hernández, Dieter Remus, Javier Trigos-Arrieta F (2001) Some open problems on topological groups. In: Martín Peinadar E et al (eds) Nuclear Groups and Lie Groups, pp 57–76. Proc. Workshop on topological groups and Lie groups (Madrid, 1999), Berlin: Heldermann
Curtis CW, Reiner I (1962) Representation Theory of Finite Groups and Associative Algebras. New York: Interscience Publishers. Reprinted 1988
M Enock JM Schwartz (1992) Kac Algebras and Duality of Locally Compact Groups Springer Berlin Heidelberg New York Occurrence Handle0805.22003
J Galindo S Hernández (2004) ArticleTitleInterpolation sets and the Bohr topology of locally compact groups Adv Math 188 51–68 Occurrence Handle1055.22003 Occurrence Handle2083092 Occurrence Handle10.1016/j.aim.2003.09.006
M Goto (1961) ArticleTitleNote on a topology of a dual space Proc Amer Math Soc 12 41–46 Occurrence Handle0132.27704 Occurrence Handle125898 Occurrence Handle10.2307/2034121
H Heyer (1970) Dualität lokalkompakter Gruppen Springer Berlin Heidelberg New York Occurrence Handle0202.14003
Heyer H (1973) Groups with Chu duality. In: Probability and Information Theory, Lect Notes Math 296: 181–215. Berlin Heidelberg New York: Springer
G Hochschild DG Mostow (1957) ArticleTitleRepresentations and representative functions of Lie groups Ann Math 66 495–542 Occurrence Handle0080.25101 Occurrence Handle98796 Occurrence Handle10.2307/1969906
G Lukács (2003) ArticleTitleOn homomorphism spaces of metrizable groups J Pure Appl Algebra 182 263–267 Occurrence Handle1025.22002 Occurrence Handle1984395 Occurrence Handle10.1016/S0022-4049(03)00028-8
W Moran (1971) ArticleTitleOn almost periodic compactifications of locally compact groups J London Math Soc 3 507–512 Occurrence Handle0229.22008 Occurrence Handle296216
D Poguntke (1972) ArticleTitleA universal property of the Takahashi quasi-dual Can J Math XXIV 530–536 Occurrence Handle301136
D Poguntke (1976) ArticleTitleZwei Klassen lokalkompakter maximal fastperiodischer Gruppen Monatsh Math 81 15–40 Occurrence Handle0322.22007 Occurrence Handle425002 Occurrence Handle10.1007/BF01473612
D Poguntke (1976) ArticleTitleChu-Dualität und zwei Klassen maximal fastperiodischer Gruppen Monatsh Math 82 31–50 Occurrence Handle0354.22006 Occurrence Handle425003 Occurrence Handle10.1007/BF01304377
D Remus FJ Trigos-Arrieta (1999) ArticleTitleThe Bohr topology of Moore groups Top Appl 97 85–98 Occurrence Handle0933.22008 Occurrence Handle1676673 Occurrence Handle10.1016/S0166-8641(98)00070-4
Riggins L (1998) On infinite groups and unitary duality. Ph.D. thesis. Cleveland, Ohio: Case Western Reserve University
DJS Robinson (1982) A Course in the Theory of Groups Springer Berlin Heidelberg New York Occurrence Handle0836.20001
DW Roeder (1970) ArticleTitleA characterization of unitary duality Trans Amer Math Soc 148 124–135 Occurrence Handle255737 Occurrence Handle10.2307/1995042
S Takahashi (1952) ArticleTitleA duality theorem for representable locally compact groups with compact commutator subgroup Tôhoku Math J 4 115–121 Occurrence Handle0047.26101
TS Wu L Riggins (1996) ArticleTitleMaximally almost periodic groups and a theorem of Glicksberg Annals NY Acad Science 806 454–464 Occurrence Handle0889.22003 Occurrence Handle1429673
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The first named author acknowledges partial financial support by the Spanish Ministry of Science (including FEDER funds), grant MTM2004-07665-C02-01; and the Generalitat Valenciana, grant GV04B-019.
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Hernández, S., Wu, TS. Some New Results on the Chu Duality of Discrete Groups. Mh Math 149, 215–232 (2006). https://doi.org/10.1007/s00605-006-0382-z
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DOI: https://doi.org/10.1007/s00605-006-0382-z