Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we study the ideals in the Fourier algebra of a locally compact group $G$which are complemented by an invariant projection. In particular we show that when $G$ 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.

References [Enhancements On Off] (What's this?)

  • 1. D.P Blecher, The standard dual of an operator space, Pacific J. Math. 153 (1992), 15-30. MR 93d:47083
  • 2. P.C. Curtis and R.J. Loy, Amenable Banach algebras, J. London Math. Soc. 40 (1989) 89-104. MR 90k:46114
  • 3. E.G. Effros and Z-J Ruan, On the abstract characterization of operator spaces, Proc. Amer. Math. Soc. 119 (1990), 579-584. MR 94g:46019
  • 4. P. Eymard, L'algebre de Fourier d'un groupe localemant compact, Bull. Soc. Math. France 92 (1964), 181-236. MR 37:4208
  • 5. B. Johnson, Cohomology in Banach Algebras, Mem. Amer. Math. Soc. 127 (1972). MR 51:11130
  • 6. -, Nonamenability of the Fourier algebra for compact groups, J. London Math. Soc. 50 (1994), 361-374. MR 95i:43001
  • 7. A. Ya. Khelemskii, Flat Banach modules and amenable algebras, Trans. Moscow Math. Soc. (1984); Amer. Math. Soc. Translations (1985), 199-224. MR 86g:46108
  • 8. -, The Homology of Banach and Topological Algebras, Mathematics and its Applications (Soviet Series) 41, Kluwer Academic Publishers, 1986. MR 92d:46178
  • 9. D.J. Newman, The non-existence of projections from $L^1({\mathbb T})$ to $H^1({\mathbb T})$, Proc. Amer. Math. Soc. 12 (1961), 98-99. MR 22:11276
  • 10. H.P. Rosenthal, Projections onto translation invariant subspaces of $L^p(G)$, Mem. Amer. Math. Soc. 63 (1966), 84pp. MR 35:2080
  • 11. Z-J Ruan, The operator amenability of $A(G)$, Amer. J. Math. 117 (1995), 1449-1476. MR 96m:43001
  • 12. W. Rudin, Projections on invariant subspaces, Proc. Amer. Math. Soc. 13 (1962), 429-432. MR 25:1460
  • 13. P.J. Wood, Complemented ideals in the Fourier algebra of a locally compact group, Proc. Amer. Math. Soc. 128 No. 2 (2000), 445-451. MR 2000c:43004
  • 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).

Similar Articles

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

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