## 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.

### MSC:

 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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### References:

 [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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