Amenability and weak amenability of the Fourier algebra. (English) Zbl 1080.22002

For a locally compact group \(G\) the Fourier algebra \(A(G)\) [P. Eymard, C. R. Acad. Sci. Paris 256, 1429–1431 (1963; Zbl 0118.11402)] and the Fourier-Stieltjes algebra \(B(G)\) [P. Eymard, Bull. Soc. Math. France 92, 181–236 (1964; Zbl 0169.46403)] are commutative Banach algebras. C. Moore [Trans. Am. Math. Soc. 166, 401–410 (1972; Zbl 0236.22010)] proved that \(G\) has an open Abelian subgroup of finite index (i.e., it has a finite number of cosets) if and only if all \(\pi \in \hat{G}\) are finite dimensional.
The authors prove in the first section for a function \(f\) on \(G\) that the homomorphism \(f(x) \mapsto f(x^{-1})\) is completely bounded if \(G\) has an Abelian subgroup of finite index. The converse result had been proved by B. Forrest and P. Wood [Indiana Univ. Math. J. 50, No. 3, 1217–1240 (2001; Zbl 1037.43005)]. It is known that if \(G\) has an abelian subgroup of finite index then \(A(G)\) is amenable [A. T.-M. Lau et al., Stud. Math. 119, No. 2, 161–178 (1996; Zbl 0858.46038)]. To prove the converse of this, the authors deal first with discrete groups; \(G\) has an Abelian subgroup of finite index if and only if the antidiagonal (i.e.,the set of pairs \(\{ x, x^{-1} \}, x \in G\)) is in the coset ring (i.e., the ring of subsets generated by all left cosets of \(G \times G\)). They use the complete boundedness of the natural endomorphism of \( A(G)\) determined by \(x \mapsto x^{-1}\) [M. Ilie and N. Spronk, J. Funct. Anal. 225, No. 2, 480–499 (2005; Zbl 1077.43004)] to extend the above to locally compact \(G\). They then show that if \(A(G)\) is amenable the anti-diagonal is in the coset ring so proving that \(G\) has an Abelian subgroup of finite index.
If \(G\) has a compact Abelian subgroup of finite index then it is obvious from the preceding theorem that \(B(G)\) is amenable. The authors prove the converse by a circuitious route: \(B(G)\) amenable \(\Rightarrow\) \(A(G)\) amenable \(\Rightarrow\) \(G\) has an Abelian subgroup \(A\) of finite index \(\Rightarrow\) \(B(G)\) amenable \(\Rightarrow\) \(B(A)\) amenable \(\Rightarrow\) the measure algebra of \(\hat{A}\) is amenable \(\Rightarrow\) \(A\) is discrete \(\Rightarrow\) \(A\) is compact. The authors believe that there should be a more direct proof using the explicit form of the coset ring given by B. Forrest et al. [J. Funct. Anal. 203, No. 1, 286–304 (2003; Zbl 1039.46042)].
A commutative Banach algebra \(\mathcal{A}\) is called weakly amenable if every continuous derivation into a commutative \(\mathcal{A}\)-module is identically zero [W. G. Bade et al., Proc. Lond. Math. Soc., III. Ser. 55, 359–377 (1987; Zbl 0634.46042)]. The authors prove that if \(G\) is locally compact with Abelian component of the identity then \(A(G)\) is weakly amenable. The authors finally develop a converse to this for groups with arbitrarily small conjugation-invariant neighbourhoods [see C. Moore, loc.cit.] and a different definition of amenability slightly stronger than weak amenability.


22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
47L50 Dual spaces of operator algebras
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[1] Albrecht, E., Dales, H. G.: Continuity of homomorphisms from C*-algebras and other Banach algebras. In: J. M. Bachar et al. (ed.s), Radical Banach Algebras and Automatic Continuity, Lectures Notes in Mathematics 975, pp. 375–396. Sprinver Verlag, 1983 · Zbl 0518.46034
[2] Bade, W. G., Curtis, Jr. P. C., Dales, H. G.: Amenability and weak amenability for Beurling and Lipschitz algebras. Proc. London Math. Soc. 55 , 359–377 (1987) · Zbl 0634.46042
[3] Brown, G., Moran, W.: Point derivations on M(G). Bull. London Math. Soc. 8, 57–64 (1976) · Zbl 0321.43003
[4] Dales, H. G., Ghahramani, F., Helemskii, A. Ya.: The amenability of measure algebras. J. London Math. Soc. 66(2), 213–226 (2002) · Zbl 1015.43002
[5] Effros, E. G., Ruan, Z.-J.: Operator Spaces. Oxford University Press, 2000
[6] Eymard, P.: L’algèbre de Fourier d’un groupe localement compact. Bull. Soc. Math. France 92 , 181–236 (1964) · Zbl 0169.46403
[7] Forrest, B. E.: Amenability and bounded approximate identities in ideals of A(G). Illinois J. Math. 34, 1–25 (1987) · Zbl 0712.43002
[8] Forrest, B. E.: Amenability and ideals in A(G). J. Austral. Math. Soc. A 53 , 143–155 (1992) · Zbl 0767.43003
[9] Forrest, B. E.: Fourier analysis on coset spaces. Rocky Mountain J. Math. 28, 173–190 (1998) · Zbl 0922.43007
[10] Forrest, B. E.: Amenability and weak amenability of the Fourier algebra. Preprint (2000) · Zbl 1080.22002
[11] Forrest, B. E., Wood, P. J.: Cohomology and the operator space structure of the Fourier algebra and its second dual. Indiana Univ. Math. J. 50, 1217–1240 (2001) · Zbl 1037.43005
[12] Forrest, B. E., Kaniuth, E., Lau, A. T.-M., Spronk, N.: Ideals with bounded approximate identities in Fourier algebras. J. Funct. Anal. 203, 286–304 (2003) · Zbl 1039.46042
[13] Grønbæk, N.: A characterization of weakly amenable Banach algebras. Studia Math. 94, 149–162 (1989)
[14] Helemskii, A. Ya.: The Homology of Banach and Topological Algebras (translated from the Russian). Kluwer Academic Publishers, 1989
[15] Hewitt, E., Ross, K. A.: Abstract Harmonic Analysis. I. Springer Verlag, 1963 · Zbl 0115.10603
[16] Host, B.: Le théorème des idempotents dans B(G). Bull. Soc. Math. France 114, 215–223 (1986) · Zbl 0606.43002
[17] Ilie, M.: Homomorphisms of Fourier Algebras and Coset Spaces of a Locally Compact Group. Ph.D. thesis, University of Alberta, 2003
[18] Ilie, M., Spronk, N.: Completely bounded homomorphisms of the Fourier algebras. J. Funct. Anal. (to appear) · Zbl 1077.43004
[19] Johnson, B. E.: Cohomology in Banach algebras. Mem. Amer. Math. Soc. 127, (1972) · Zbl 0256.18014
[20] Johnson, B. E.: Approximate diagonals and cohomology of certain annihilator Banach algebras. Amer. J. Math. 94, 685–698 (1972) · Zbl 0246.46040
[21] Johnson, B. E.: Non-amenability of the Fourier algebra of a compact group. J. London Math. Soc. 50(2), 361–374 (1994) · Zbl 0829.43004
[22] Lau, A. T.-M., Loy, R. J., Willis, G. A.: Amenability of Banach and C*-algebras on locally compact groups. Studia Math. 119, 161–178 (1996) · Zbl 0858.46038
[23] Leptin, H.: Sur l’algèbre de Fourier d’un groupe localement compact. C. R. Acad. Sci. Paris, Sér. A 266, 1180–1182 (1968) · Zbl 0169.46501
[24] Losert, V.: On tensor products of Fourier algebras. Arch. Math. (Basel) 43, 370–372 (1984) · Zbl 0587.43001
[25] Moore, C. C.: Groups with finite dimensional irreducible representations. Trans. Amer. Math. Soc. 166, 401–410 (1972) · Zbl 0236.22010
[26] Palmer, T. W.: Banach Algebras and the General Theory of *-Algebras, II. Cambridge University Press, 2001 · Zbl 0983.46040
[27] Paterson, A. L. T.: Amenability. American Mathematical Society, 1988
[28] Ruan, Z.-J.: The operator amenability of A(G). Amer. J. Math. 117, 1449–1474 (1995) · Zbl 0842.43004
[29] Rudin, W.: Fourier Analysis on Groups. Wiley-Interscience, 1990 · Zbl 0698.43001
[30] Runde, V.: Lectures on Amenability. Lecture Notes in Mathematics 1774, Springer Verlag, 2002 · Zbl 0999.46022
[31] Runde, V.: Operator Figà-Talamanca–Herz algebras. Studia Math. 155, 153–170 (2003) · Zbl 1032.47048
[32] Runde, V.: (Non-)amenability of Fourier and Fourier–Stieltjes algebras. Preprint (2002) · Zbl 0999.46022
[33] Runde, V., Spronk, N.: Operator amenability of Fourier-Stieltjes algebras. Math. Proc. Cambridge Phil. Soc. 136, 675–686 (2004) · Zbl 1052.43003
[34] Spronk, N.: Operator weak amenability of the Fourier algebra. Proc. Amer. Math. Soc. 130, 3609–3617 (2002) · Zbl 1006.46040
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