Complemented ideals in the Fourier algebra of a locally compact group

Wood, Peter

doi:10.1090/S0002-9939-99-04989-8

Complemented ideals in the Fourier algebra of a locally compact group
HTML articles powered by AMS MathViewer

by Peter J. Wood

Proc. Amer. Math. Soc. 128 (2000), 445-451

DOI: https://doi.org/10.1090/S0002-9939-99-04989-8

Published electronically: October 12, 1999

PDF | Request permission

Abstract:

In this paper we provide a necessary condition for a closed ideal in the Fourier algebra of a locally compact amenable group to be completely complemented. The classification of completely complemented ideals is completed in the case of an amenable discrete group. We also investigate the ideals possessing a bounded approximate identity.

References

Dale E. Alspach and Alec Matheson, Projections onto translation-invariant subspaces of $L_{1}(\textbf {R})$, Trans. Amer. Math. Soc. 277 (1983), no. 2, 815–823. MR 694390, DOI 10.1090/S0002-9947-1983-0694390-7
David P. Blecher, A completely bounded characterization of operator algebras, Math. Ann. 303 (1995), no. 2, 227–239. MR 1348798, DOI 10.1007/BF01460988
David P. Blecher, The standard dual of an operator space, Pacific J. Math. 153 (1992), no. 1, 15–30. MR 1145913
David P. Blecher, Commutativity in operator algebras, Proc. Amer. Math. Soc. 109 (1990), no. 3, 709–715. MR 1009985, DOI 10.1090/S0002-9939-1990-1009985-X
David P. Blecher and Vern I. Paulsen, Tensor products of operator spaces, J. Funct. Anal. 99 (1991), no. 2, 262–292. MR 1121615, DOI 10.1016/0022-1236(91)90042-4
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, DOI 10.1112/jlms/s2-40.1.89
Edward G. Effros and Zhong-Jin Ruan, On matricially normed spaces, Pacific J. Math. 132 (1988), no. 2, 243–264. MR 934168
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, DOI 10.1090/S0002-9939-1993-1163332-4
Pierre Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236 (French). MR 228628
Brian Forrest, Amenability and bounded approximate identities in ideals of $A(G)$, Illinois J. Math. 34 (1990), no. 1, 1–25. MR 1031879
Barry Edward Johnson, Cohomology in Banach algebras, Memoirs of the American Mathematical Society, No. 127, American Mathematical Society, Providence, R.I., 1972. MR 0374934
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, DOI 10.1112/jlms/50.2.361
B. Host, Le théorème des idempotents dans $B(G)$, Bull. Soc. Math. France 114 (1986), no. 2, 215–223 (French, with English summary). MR 860817
A. Ya. Khelemskiĭ, Flat Banach modules and amenable algebras, Trudy Moskov. Mat. Obshch. 47 (1984), 179–218, 247 (Russian). MR 774950
Michael Leinert, Faltungsoperatoren auf gewissen diskreten Gruppen, Studia Math. 52 (1974), 149–158 (German). MR 355480, DOI 10.4064/sm-52-2-149-158
Lui, T-S, van Rooij A. and Wang, J.K., Projections and approximate identities for ideals in group algebras, Trans. Amer. Math. Soc. 175 (1973) 469-482
Vern I. Paulsen, Representations of function algebras, abstract operator spaces, and Banach space geometry, J. Funct. Anal. 109 (1992), no. 1, 113–129. MR 1183607, DOI 10.1016/0022-1236(92)90014-A
Haskell P. Rosenthal, Projections onto translation-invariant subspaces of $L^{p}(G)$, Mem. Amer. Math. Soc. 63 (1966), 84. MR 211198
Walter Rudin, Projections on invariant subspaces, Proc. Amer. Math. Soc. 13 (1962), 429–432. MR 138012, DOI 10.1090/S0002-9939-1962-0138012-4
Zhong-Jin Ruan, The operator amenability of $A(G)$, Amer. J. Math. 117 (1995), no. 6, 1449–1474. MR 1363075, DOI 10.2307/2375026
Ruan, Z-J., and Xu, G., Splitting properties of operator bimodules and operator amenability of Kac algebras (1996) preprint