Weak amenability of Fourier algebras and local synthesis of the anti-diagonal

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• 作者：Hun Hee Lee^a ; ¹ ; ^{hunheelee@snu.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Jean Ludwig^b ; ² ; ^{ludwig@univ-metz.fr" class="auth_mail" title="E-mail the corresponding author} ; Ebrahim Samei^c ; ³ ; ^{samei@math.usask.ca" class="auth_mail" title="E-mail the corresponding author} ; Nico Spronk^d ; ⁴ ; ^{nspronk@uwaterloo.ca" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 43A30 ; secondary ; 43A45 ; 43A80 ; 22E15 ; 22D35 ; 46H25 ; 46J40
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：9 April 2016
• 年：2016
• 卷：292
• 期：Complete
• 页码：11-41
• 全文大小：562 K

文摘

We show that for a connected Lie group G  , its Fourier algebra A(G)$A(G)$ is weakly amenable only if G   is abelian. Our main new idea is to show that weak amenability of A(G)$A(G)$ implies that the anti-diagonal, ${\check{\mathrm{\Delta }}}_G=\{(g,g^{-1}):g\in G\}$, is a set of local synthesis for A(G×G)$A(G\times G)$. We then show that this cannot happen if G is non-abelian. We conclude for a locally compact group G  , that A(G)$A(G)$ can be weakly amenable only if it contains no closed connected non-abelian Lie subgroups. In particular, for a Lie group G  , A(G)$A(G)$ is weakly amenable if and only if its connected component of the identity 30ef4839d" title="Click to view the MathML source">G_e$G_e$ is abelian.