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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:
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