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Open subgroups and the centre problem for the Fourier algebra


Author: Zhiguo Hu
Journal: Proc. Amer. Math. Soc. 134 (2006), 3085-3095
MSC (2000): Primary 22D25, 43A30
DOI: https://doi.org/10.1090/S0002-9939-06-08334-1
Published electronically: May 5, 2006
MathSciNet review: 2231636
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Abstract: Let $ A(G)$ be the Fourier algebra of a locally compact group and $ UCB(\hat{G})$ the $ C^*$-algebra of uniformly continuous linear functionals on $ A(G)$. We study how the centre problem for the algebra $ UCB(\hat{G})^*$ (resp. $ A(G)^{**}$) is related to the centre problem for the algebras $ UCB(\hat{H})^*$ (resp. $ A(H)^{**}$) of $ \sigma$-compact open subgroups $ H$ of $ G$. We extend some results of Lau-Losert on the centres of $ UCB(\hat{G})^*$ and $ A(G)^{**}$.


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  • 1. R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839-848. MR 0045941 (13:659f)
  • 2. P. Eymard, L'algèbra de Fourier d'un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181-236. MR 0228628 (37:4208)
  • 3. E. E. Granirer, Weakly almost periodic and uniformly continuous functionals on the Fourier algebra of any locally compact group, Trans. Amer. Math. Soc. 189 (1974), 371-382. MR 0336241 (49:1017)
  • 4. E. E. Granirer, Density theorems for some linear subspaces and some $ C^*$-algebras of $ VN(G)$, Istituto Nazionale di Alta Mathematica, Symposia Mathematica 22 (1977), 61-70. MR 0487287 (58:6935)
  • 5. M. Grosser and V. Losert, The norm-strict bidual of a Banach algebra and the dual of $ C_u(G)$, Manuscript Math. 45 (1984), 127-146. MR 0724731 (86b:46073)
  • 6. C. Herz, Harmonic synthesis for subgroups, Annales de l'Institut Fourier (Grenoble) 23 (1973), 91-123. MR 0355482 (50:7956)
  • 7. E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I, Springer-Verlag, New York, 1979. MR 0551496 (81k:43001)
  • 8. Z. Hu, Open subgroups of $ G$ and almost periodic functionals on $ A(G)$, Proc. Amer. Math. Soc. 128 (2000), 2473-2478. MR 1662249 (2000k:22009)
  • 9. Z. Hu, Inductive extreme non-Arens regularity of the Fourier algebra $ A(G)$, Studia Math. 151 (2002), 247-264. MR 1917836 (2003f:46069)
  • 10. Z. Hu, Maximally decomposable von Neumann algebras on locally compact groups and duality, Houston J. Math. 31 (2005), 857-881. MR 2148807
  • 11. Z. Hu and M. Neufang, Decomposability of von Neumann algebras and Mazur property of higher level, Canad. J. Math, to appear.
  • 12. N. Isik, J. S. Pym and A. Ülger, The second dual of the group algebra of a compact group, J. London Math. Soc. (2) 35 (1987), 135-148. MR 0871771 (88f:43012)
  • 13. E. Kaniuth, A. T. Lau and G. Schlichting, Lebesgue type decomposition of subspaces of Fourier-Stieltjes algebras, Trans. Amer. Math. Soc. 355 (2003), 1467-1490. MR 1946400 (2004c:43004)
  • 14. A. T. Lau, Uniformly continuous functionals of the Fourier algebra of any locally compact group, Trans. Amer. Math. Soc. 251 (1979), 39-59. MR 0531968 (80m:43009)
  • 15. A. T. Lau, Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups, Math. Proc. Cambridge Philos. Soc. 99 (1986), 273-283. MR 0817669 (87i:43001)
  • 16. A. T. Lau and V. Losert, On the second conjugate algebra of a locally compact group, J. London Math. Soc. 37 (1988), 464-470. MR 0939122 (89e:43007)
  • 17. A. T. Lau and V. Losert, The $ C^*$-algebra generated by operators with compact support on a locally compact group, J. Funct. Anal. 112 (1993), 1-30. MR 1207935 (94d:22005)
  • 18. A. T. Lau and V. Losert, The centre of the second conjugate algebra of the Fourier algebra for infinite product of groups, Math. Proc. Cambridge Philos. Soc., 138 (2005), 27-39. MR 2127225 (2006c:43003)
  • 19. V. Losert, The centre of the bidual of Fourier algebras (discrete groups), preprint.
  • 20. A. Ülger, Central elements of $ A^{**}$ for certain Banach algebras $ A$ without bounded approximate identities, Glasgow Math. J. 41(1999), 369-377. MR 1720442 (2001b:46082)

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Additional Information

Zhiguo Hu
Affiliation: Department of Mathematics and Statistics, University of Windsor, Windsor, Ontario, Canada N9B 3P4
Email: zhiguohu@uwindsor.ca

DOI: https://doi.org/10.1090/S0002-9939-06-08334-1
Keywords: Fourier algebra, reduced Fourier-Stieltjes algebra
Received by editor(s): January 11, 2005
Received by editor(s) in revised form: May 5, 2005
Published electronically: May 5, 2006
Additional Notes: This research was supported by an NSERC grant.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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