Amenability and the structure of the algebras 𝐴_{𝑝}(𝐺)

Forrest, Brian

doi:10.1090/S0002-9947-1994-1181182-5

Amenability and the structure of the algebras $A\sb p(G)$

Author: Brian Forrest
Journal: Trans. Amer. Math. Soc. 343 (1994), 233-243
MSC: Primary 43A07; Secondary 43A15, 46J99
DOI: https://doi.org/10.1090/S0002-9947-1994-1181182-5
MathSciNet review: 1181182
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A number of characterizations are given of the class of amenable locally compact groups in terms of the ideal structure of the algebras ${A_p}(G)$ . An almost connected group is amenable if and only if for some $1 < p < \infty$ and some closed ideal I of ${A_p}(G)$ , I has a bounded approximate identity. Furthermore, G is amenable if and only if every derivation of ${A_p}(G)$ into a Banach ${A_p}(G)$ -bimodule is continuous.

[1] W. G. Bade and P. C. Curtis Jr., Prime ideals and automatic continuity problems for Banach algebras, J. Funct. Anal. 29 (1978), no. 1, 88–103. MR 499936, https://doi.org/10.1016/0022-1236(78)90048-4
[2] C. P. Chen and P. J. Cohen, Ideals of finite codimension in commutative Banach algebras, preprint.
[3] Michael Cowling, An application of Littlewood-Paley theory in harmonic analysis, Math. Ann. 241 (1979), no. 1, 83–96. MR 531153, https://doi.org/10.1007/BF01406711
[4] Michael Cowling and Paul Rodway, Restrictions of certain function spaces to closed subgroups of locally compact groups, Pacific J. Math. 80 (1979), no. 1, 91–104. MR 534697
[5] 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, https://doi.org/10.1112/jlms/s2-40.1.89
[6] Antoine Derighetti, Convoluteurs et projecteurs, Harmonic analysis (Luxembourg, 1987) Lecture Notes in Math., vol. 1359, Springer, Berlin, 1988, pp. 142–158 (French). MR 974311, https://doi.org/10.1007/BFb0086595
[7] Pierre Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236 (French). MR 0228628
[8] Hans G. Feichtinger, Colin C. Graham, and Eric H. Lakien, Nonfactorization in commutative, weakly selfadjoint Banach algebras, Pacific J. Math. 80 (1979), no. 1, 117–125. MR 534699
[9] Brian Forrest, Amenability and derivations of the Fourier algebra, Proc. Amer. Math. Soc. 104 (1988), no. 2, 437–442. MR 931730, https://doi.org/10.1090/S0002-9939-1988-0931730-5
[10] Brian Forrest, Amenability and bounded approximate identities in ideals of 𝐴(𝐺), Illinois J. Math. 34 (1990), no. 1, 1–25. MR 1031879
[11] Brian Forrest, Amenability and ideals in 𝐴(𝐺), J. Austral. Math. Soc. Ser. A 53 (1992), no. 2, 143–155. MR 1175708
[12] John E. Gilbert, On projections of 𝐿^{∞}(𝐺) onto translation-invariant subspaces, Proc. London Math. Soc. (3) 19 (1969), 69–88. MR 0244705, https://doi.org/10.1112/plms/s3-19.1.69
[13] Andrew M. Gleason, A characterization of maximal ideals, J. Analyse Math. 19 (1967), 171–172. MR 0213878, https://doi.org/10.1007/BF02788714
[14] Carl Herz, Synthèse spectrale pour les sous-groupes par rapport aux algèbres 𝐴_{𝑝}, C. R. Acad. Sci. Paris Sér. A-B 271 (1970), A316–A318 (French). MR 0273429
[15] Carl Herz, The theory of 𝑝-spaces with an application to convolution operators., Trans. Amer. Math. Soc. 154 (1971), 69–82. MR 0272952, https://doi.org/10.1090/S0002-9947-1971-0272952-0
[16] Carl Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 3, 91–123 (English, with French summary). MR 0355482
[17] Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496
[18] 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
[19] Barry Edward Johnson, Cohomology in Banach algebras, American Mathematical Society, Providence, R.I., 1972. Memoirs of the American Mathematical Society, No. 127. MR 0374934
[20] J.-P. Kahane and W. Żelazko, A characterization of maximal ideals in commutative Banach algebras, Studia Math. 29 (1968), 339–343. MR 0226408, https://doi.org/10.4064/sm-29-3-339-343
[21] Anthony To Ming Lau and Viktor Losert, Weak*-closed complemented invariant subspaces of 𝐿_{∞}(𝐺) and amenable locally compact groups, Pacific J. Math. 123 (1986), no. 1, 149–159. MR 834144
[22] T. W. Palmer, Classes of nonabelian, noncompact, locally compact groups, Rocky Mountain J. Math. 8 (1978), no. 4, 683–741. MR 513952, https://doi.org/10.1216/RMJ-1978-8-4-683
[23] A. L. T. Paterson, Amenability, Amer. Math. Soc., Providence, RI, 1988.
[24] Jean-Paul Pier, Amenable locally compact groups, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 767264
[25] N. V. Rao, Closed ideals of finite codimension in regular selfadjoint Banach algebras, J. Funct. Anal. 82 (1989), no. 2, 237–258. MR 987293, https://doi.org/10.1016/0022-1236(89)90070-0
[26] C. Robert Warner and Robert Whitley, Ideals of finite codimension in 𝐶[0,1] and 𝐿¹(𝑅), Proc. Amer. Math. Soc. 76 (1979), no. 2, 263–267. MR 537085, https://doi.org/10.1090/S0002-9939-1979-0537085-7