Best bounds for approximate identities in ideals of the Fourier algebra vanishing on subgroups

Authors:
Brian Forrest and Nicolaas Spronk

Journal:
Proc. Amer. Math. Soc. **134** (2006), 111-116

MSC (2000):
Primary 43A30, 46J20; Secondary 46L07, 43A07

Published electronically:
August 15, 2005

MathSciNet review:
2170550

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we show that if is an amenable locally compact group and if is a closed subgroup, then the ideal has an approximate identity of norm If is not open, this bound is the best possible.

**1.**David P. Blecher,*The standard dual of an operator space*, Pacific J. Math.**153**(1992), no. 1, 15–30. MR**1145913****2.**Jacques Delaporte and Antoine Derighetti,*On ideals of 𝐴_{𝑝} with bounded approximate units and certain conditional expectations*, J. London Math. Soc. (2)**47**(1993), no. 3, 497–506. MR**1214911**, 10.1112/jlms/s2-47.3.497**3.**Jacques Delaporte and Antoine Derighetti,*Best bounds for the approximate units for certain ideals of 𝐿¹(𝐺) and of 𝐴_{𝑝}(𝐺)*, Proc. Amer. Math. Soc.**124**(1996), no. 4, 1159–1169. MR**1301019**, 10.1090/S0002-9939-96-03130-9**4.**Pierre Eymard,*L’algèbre de Fourier d’un groupe localement compact*, Bull. Soc. Math. France**92**(1964), 181–236 (French). MR**0228628****5.**Brian Forrest,*Amenability and bounded approximate identities in ideals of 𝐴(𝐺)*, Illinois J. Math.**34**(1990), no. 1, 1–25. MR**1031879****6.**B. Forrest, E. Kaniuth, A. T. Lau, and N. Spronk,*Ideals with bounded approximate identities in Fourier algebras*, J. Funct. Anal.**203**(2003), no. 1, 286–304. MR**1996874**, 10.1016/S0022-1236(02)00121-0**7.**Carl Herz,*The theory of 𝑝-spaces with an application to convolution operators.*, Trans. Amer. Math. Soc.**154**(1971), 69–82. MR**0272952**, 10.1090/S0002-9947-1971-0272952-0**8.**B. E. Johnson,*Non-amenability of the Fourier algebra of a compact group*, J. London Math. Soc. (2)**50**(1994), no. 2, 361–374. MR**1291743**, 10.1112/jlms/50.2.361**9.**Eberhard Kaniuth and Anthony T. Lau,*A separation property of positive definite functions on locally compact groups and applications to Fourier algebras*, J. Funct. Anal.**175**(2000), no. 1, 89–110. MR**1774852**, 10.1006/jfan.2000.3612**10.**E. Kaniuth and A.T. Lau,*On a separation property of positive definite functions on locally compact groups*. (preprint).**11.**John L. Kelley,*General topology*, Springer-Verlag, New York-Berlin, 1975. Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.]; Graduate Texts in Mathematics, No. 27. MR**0370454****12.**Zhong-Jin Ruan,*The operator amenability of 𝐴(𝐺)*, Amer. J. Math.**117**(1995), no. 6, 1449–1474. MR**1363075**, 10.2307/2375026**13.**Volker Runde,*Lectures on amenability*, Lecture Notes in Mathematics, vol. 1774, Springer-Verlag, Berlin, 2002. MR**1874893****14.**Peter J. Wood,*Complemented ideals in the Fourier algebra of a locally compact group*, Proc. Amer. Math. Soc.**128**(2000), no. 2, 445–451. MR**1616589**, 10.1090/S0002-9939-99-04989-8

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
43A30,
46J20,
46L07,
43A07

Retrieve articles in all journals with MSC (2000): 43A30, 46J20, 46L07, 43A07

Additional Information

**Brian Forrest**

Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Email:
beforres@math.uwaterloo.ca

**Nicolaas Spronk**

Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Email:
nspronk@uwaterloo.ca

DOI:
http://dx.doi.org/10.1090/S0002-9939-05-08205-5

Keywords:
Fourier algebra,
ideal,
bounded approximate identity,
operator space

Received by editor(s):
December 3, 2003

Published electronically:
August 15, 2005

Additional Notes:
The first author was supported in part by a grant from NSERC. The second author was a visiting assistant professor at Texas A&M University when this work was completed and was supported in part by an NSERC PDF

Communicated by:
David R. Larson

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.