Invariant complementation and projectivity in the Fourier algebra

Author:
Peter J. Wood

Journal:
Proc. Amer. Math. Soc. **131** (2003), 1881-1890

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

Published electronically:
November 4, 2002

MathSciNet review:
1955277

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we study the ideals in the Fourier algebra of a locally compact group which are complemented by an invariant projection. In particular we show that when is discrete, every ideal which is complemented by a completely bounded projection must be invariantly complemented. Perhaps surprisingly, this result does not depend of the amenability of the group or the algebra, but instead relies on the operator biprojectivity of the Fourier algebra for a discrete group.

**1.**David P. Blecher,*The standard dual of an operator space*, Pacific J. Math.**153**(1992), no. 1, 15–30. MR**1145913****2.**P. C. Curtis Jr. and R. J. Loy,*The structure of amenable Banach algebras*, J. London Math. Soc. (2)**40**(1989), no. 1, 89–104. MR**1028916**, 10.1112/jlms/s2-40.1.89**3.**Edward G. Effros and Zhong-Jin Ruan,*On the abstract characterization of operator spaces*, Proc. Amer. Math. Soc.**119**(1993), no. 2, 579–584. MR**1163332**, 10.1090/S0002-9939-1993-1163332-4**4.**Pierre Eymard,*L’algèbre de Fourier d’un groupe localement compact*, Bull. Soc. Math. France**92**(1964), 181–236 (French). MR**0228628****5.**Barry Edward Johnson,*Cohomology in Banach algebras*, American Mathematical Society, Providence, R.I., 1972. Memoirs of the American Mathematical Society, No. 127. MR**0374934****6.**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**7.**A. Ya. Khelemskiĭ,*Flat Banach modules and amenable algebras*, Trudy Moskov. Mat. Obshch.**47**(1984), 179–218, 247 (Russian). MR**774950****8.**A. Ya. Helemskii,*The homology of Banach and topological algebras*, Mathematics and its Applications (Soviet Series), vol. 41, Kluwer Academic Publishers Group, Dordrecht, 1989. Translated from the Russian by Alan West. MR**1093462****9.**D. J. Newman,*The nonexistence of projections from 𝐿¹ to 𝐻¹*, Proc. Amer. Math. Soc.**12**(1961), 98–99. MR**0120524**, 10.1090/S0002-9939-1961-0120524-X**10.**Haskell P. Rosenthal,*Projections onto translation-invariant subspaces of 𝐿^{𝑝}(𝐺)*, Mem. Amer. Math. Soc. No.**63**(1966), 84. MR**0211198****11.**Zhong-Jin Ruan,*The operator amenability of 𝐴(𝐺)*, Amer. J. Math.**117**(1995), no. 6, 1449–1474. MR**1363075**, 10.2307/2375026**12.**Walter Rudin,*Projections on invariant subspaces*, Proc. Amer. Math. Soc.**13**(1962), 429–432. MR**0138012**, 10.1090/S0002-9939-1962-0138012-4**13.**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**14.**-,*Homological Algebra in Operator Spaces with Applications to Harmonic Analysis*, Ph.D. Thesis, University of Waterloo, 1999.**15.**-,*The operator biprojectivity of the Fourier algebra*, Can. J. Math (to appear).

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

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

Additional Information

**Peter J. Wood**

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

Email:
pwood@uwaterloo.ca

DOI:
http://dx.doi.org/10.1090/S0002-9939-02-06742-4

Keywords:
Fourier algebra,
operator space,
projective,
complemented ideals

Received by editor(s):
February 15, 2001

Received by editor(s) in revised form:
February 8, 2002

Published electronically:
November 4, 2002

Communicated by:
Andreas Seeger

Article copyright:
© Copyright 2002
American Mathematical Society