Abstract
We show that for a connected Lie group G, its Fourier algebra A(G) is weakly amenable only if G is abelian. Our main new idea is to show that weak amenability of A(G) implies that the anti-diagonal, ΔˇG={(g,g−1):g∈G}, is a set of local synthesis for A(G×G). We then show that this cannot happen if G is non-abelian. We conclude for a locally compact group G, that A(G) can be weakly amenable only if it contains no closed connected non-abelian Lie subgroups. In particular, for a Lie group G, A(G) is weakly amenable if and only if its connected component of the identity Ge is abelian.
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