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

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]**Erik Bédos,*On the 𝐶*-algebra generated by the left regular representation of a locally compact group*, Proc. Amer. Math. Soc.**120**(1994), no. 2, 603–608. MR**1181157**, 10.1090/S0002-9939-1994-1181157-1**[2]**I. M. Gali,*Positive-definite functions on direct product (sum) of locally compact abelian groups*, Proc. Pakistan Acad. Sci.**26**(1989), no. 2, 149–157. MR**1018827****[3]**Ching Chou,*Almost periodic operators in 𝑉𝑁(𝐺)*, Trans. Amer. Math. Soc.**317**(1990), no. 1, 229–253. MR**943301**, 10.1090/S0002-9947-1990-0943301-9**[4]**Charles F. Dunkl and Donald E. Ramirez,*Weakly almost periodic functionals on the Fourier algebra*, Trans. Amer. Math. Soc.**185**(1973), 501–514. MR**0372531**, 10.1090/S0002-9947-1973-0372531-2**[5]**Pierre Eymard,*L’algèbre de Fourier d’un groupe localement compact*, Bull. Soc. Math. France**92**(1964), 181–236 (French). MR**0228628****[6]**Brian Forrest,*Arens regularity and discrete groups*, Pacific J. Math.**151**(1991), no. 2, 217–227. MR**1132386****[7]**Edmond 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**, 10.1090/S0002-9947-1974-0336241-0**[8]**Edmond E. Granirer,*Density theorems for some linear subspaces and some 𝐶*-subalgebras of 𝑉𝑁(𝐺)*, Symposia Mathematica, Vol. XXII (Convegno sull’Analisi Armonica e Spazi di Funzioni su Gruppi Localmente Compatti, INDAM, Rome, 1976) Academic Press, London, 1977, pp. 61–70. MR**0487287****[9]**Anthony To Ming Lau,*Uniformly continuous functionals on the Fourier algebra of any locally compact group*, Trans. Amer. Math. Soc.**251**(1979), 39–59. MR**531968**, 10.1090/S0002-9947-1979-0531968-4**[10]**Harald Rindler,*On weak containment properties*, Proc. Amer. Math. Soc.**114**(1992), no. 2, 561–563. MR**1057960**, 10.1090/S0002-9939-1992-1057960-3**[11]**A. Ülger,*Some results about the spectrum of commutative Banach algebras under the weak topology and applications*, Monatsh. Math.**121**(1996), no. 4, 353–379. MR**1389676**, 10.1007/BF01308725

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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