Open subgroups of and almost periodic functionals on

Author:
Zhiguo Hu

Journal:
Proc. Amer. Math. Soc. **128** (2000), 2473-2478

MSC (1991):
Primary 22D25, 43A30

DOI:
https://doi.org/10.1090/S0002-9939-00-05299-0

Published electronically:
February 25, 2000

MathSciNet review:
1662249

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a locally compact group and let denote the -algebra generated by left translation operators on . Let and be the spaces of almost periodic and weakly almost periodic functionals on the Fourier algebra , respectively. It is shown that if contains an open abelian subgroup, then (1) if and only if is norm dense in ; (2) is a -algebra if is norm dense in , where denotes the set of elements in with compact support. In particular, for any amenable locally compact group which contains an open abelian subgroup, has the *dual Bohr approximation property* and is a -algebra.

**[1]**E. Bédos,*On the**-algebra generated by the left translation of a locally compact group*, Proc. Amer. Math. Soc.**120**(1994), 603-608. MR**94d:22004****[2]**M. Bekka, E. Kaniuth, A. T. Lau, and G. Schlichting,*On**-algebras associated with locally compact groups*, Proc. Amer. Math. Soc.**124**(1996), 3151-3158. MR**90m:22010****[3]**C. Chou,*Almost periodic operators in*, Trans. Amer. Math. Soc.**317**(1990), 229-253. MR**90d:43004****[4]**C. F. Dunkl and D. E. Ramirez,*Weakly almost periodic functionals on the Fourier algebra*, Trans. Amer. Math. Soc.**185**(1973), 501-514. MR**51:8738****[5]**P. Eymard,*L'algèbra de Fourier d'un groupe localement compact*, Bull. Soc. Math. France**92**(1964), 181-236. MR**37:4208****[6]**B. Forrest,*Arens regularity and discrete groups*, Pacific J. Math.**151**(1991), 217-227. MR**93c:43001****[7]**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**49:1017****[8]**E. E. Granirer,*Density theorems for some linear subspaces and some**-subalgebras of*, Sympos. Mat., vol. 22, Istit. Nazionale di Alta Mat., 1977, 61-70. MR**58:6935****[9]**A. T. Lau,*Uniformly continuous functionals on the Fourier algebra of any locally compact group*, Trans. Amer. Math. Soc.**251**(1979), 39-59. MR**80m:43009****[10]**H. Rindler,*On weak containment properties*, Proc. Amer. Math. Soc.**114**(1992), 561-563. MR**92e:22011****[11]**A. Ülger,*Some results about the spectrum of commutative Banach algebras under the weak topology and applications*, Mh. Math.**121**(1996), 353-378. MR**98a:46058**

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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/S0002-9939-00-05299-0

Keywords:
Locally compact groups,
left regular representation,
Fourier and Fourier-Stieltjes algebras,
almost periodic functionals,
weakly almost periodic functionals,
amenable groups

Received by editor(s):
December 3, 1997

Received by editor(s) in revised form:
September 8, 1998

Published electronically:
February 25, 2000

Additional Notes:
This research was supported by an NSERC grant

Communicated by:
Dale Alspach

Article copyright:
© Copyright 2000
American Mathematical Society