Weak amenability for Fourier algebras of 1-connected nilpotent Lie groups
Weak amenability for Fourier algebras of 1-connected nilpotent Lie groups
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DOI：
10.1016/j.jfa.2015.02.014
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发表时间：
2014-05
影响因子：
1.7
通讯作者：
Yemon Choi;M. Ghandehari
中科院分区：
文献类型：
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作者：
Yemon Choi;M. Ghandehari
Forrest和Runde（2005）提出的一个猜想的特例[10]断言每个非阿贝尔连通李群的傅里叶代数不弱可服从；这已经知道在非阿贝尔紧的情况下成立，由Johnson(1994)[13]和Plymen（未发表的注释）的早期工作。在最近的工作中（Choi和Ghandehari， 2014[4]），作者验证了这一猜想对于真实的a x+ b群，因此，通过结构理论，对于任何半简单李群。本文通过将问题简化到Heisenberg群的情况，验证了所有1连通非阿贝尔幂零李群的猜想。在我们之前的文章中，我们在一个稠密的子代数上构造了一个显式的非零导数，然后用调和分析证明了它是有界的。在此过程中，我们使用Schrödinger表示的已知融合规则来给出该群的“对偶卷积”作为一种扭曲的算子值卷积的具体实现。我们还给出了一些可解群的部分结果，进一步证明了一般猜想。
A special case of a conjecture raised by Forrest and Runde (2005)[10] asserts that the Fourier algebra of every non-abelian connected Lie group fails to be weakly amenable; this was already known to hold in the non-abelian compact cases, by earlier work of Johnson (1994)[13] and Plymen (unpublished note). In recent work (Choi and Ghandehari, 2014 [4]) the authors verified this conjecture for the real a x+ b group and hence, by structure theory, for any semisimple Lie group. In this paper we verify the conjecture for all 1-connected, non-abelian nilpotent Lie groups, by reducing the problem to the case of the Heisenberg group. As in our previous paper, an explicit non-zero derivation is constructed on a dense subalgebra, and then shown to be bounded using harmonic analysis. En route we use the known fusion rules for Schrödinger representations to give a concrete realization of the “dual convolution” for this group as a kind of twisted, operator-valued convolution. We also give some partial results for solvable groups which give further evidence to support the general conjecture.