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Weak and cyclic amenability for Fourier algebras of connected Lie groups. (English) Zbl 1298.43004
A commutative Banach algebra \(A\) is weakly amenable if there is no non-zero continuous derivation from \(A\) into any commutative Banach \(A\)-bimodule. B. E. Johnson showed in [J. Lond. Math. Soc., II. Ser. 50, No. 2, 361–374 (1994; Zbl 0829.43004)] that the Fourier algebra \(A(SO(3))\) of the rotation group \(SO(3)\) is not weakly amenable. Johnson’s construction was used by R. J. Plymen [“Fourier algebra of a compact Lie group”, arXiv:math/0104018] to show that \(A(G)\) is not weakly amenable for each non-abelian compact connected Lie group. This was extended by B. E. Forrest et al. [Indiana Univ. Math. J. 58, No. 3, 1379–1394 (2009; Zbl 1189.43004)] who showed that the Fourier algebra \(A(G)\) of a compact group \(G\) is weakly amenably if and only if the connected component of the identity in \(G\) is abelian. The paper under review considers non-compact groups. The authors show that \(A(G)\) is not weakly amenable in the case where \(G\) is the so-called real \(ax+b\) group. The approach is based on the orthogonality relations for the coefficient functions of the irreducible representations of that group. By combining the above mentioned result with some structure theory, the authors are able to prove that \(A(G)\) is not weakly amenable for each semisimple connected Lie group \(G\).

MSC:
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
22E15 General properties and structure of real Lie groups
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