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Non-amenability of the Fourier algebra of a compact group. (English) Zbl 0829.43004

The author pursues his investigations on the concept of amenable Banach algebras introduced by himself. The Banach algebra \(A\) is said to be [\(M\)-] amenable \([M > 0]\) if it admits an approximate diagonal, i.e., a bounded net \((t_\alpha)\) in \(A\) such that \(t_\alpha a- at_\alpha \to 0\) and \(\pi(t_{\alpha})a \to a\) for all \(a \in A\), \(\pi\) being the product map of \(A \widehat {\otimes} A\) into \(A\) [and \(|t_\alpha|\leq M\)]. In case \(G\) is an amenable group, its Fourier algebra \(A(G) = L^1(\widehat{G})\) is 1-amenable. The present article concentrates on compact groups \(G\) and the corresponding Fourier algebras \(A(G)\). If \(G\) is a finite group, \(A(G)\) is shown to be \({\sum_\pi {d_\pi}^3 \over \sum_\pi {d_\pi}^2}\)-amenable, where \(d_\pi\) denotes the dimension of the continuous representation \(\pi\) of \(G\). An arbitrary compact group \(G\) for which \((d_\pi)\) is bounded admits an amenable Fourier algebra \(A(G)\). Various special synthesis properties are established. The last section of the paper deals with the related notion of weak amenability; the condition is satisfied for compact totally disconnected groups.

MSC:

43A07 Means on groups, semigroups, etc.; amenable groups
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