A note on the density theorem for projective unitary representations
A note on the density theorem for projective unitary representations
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DOI：
10.1090/proc/13358
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
D. Han
中科院分区：
文献类型：
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作者：
D. Han
众所周知，当且仅当相关晶格满足Beurling密度条件时，Gabor表示允许一个坐标系生成器，而Beurling密度条件反过来又可以表征为相关von Neumann代数的“迹条件”。碰巧，对于任何可计数群的射影酉表示允许一个坐标系向量，这个迹路条件也是必需的。然而，对于一般的表示，特别是当Gabor表示被限制在适当的时频不变子空间时，它不再是充分的。在这篇简短的笔记中，我们证明了这个条件对于一大类射影酉表示也是充分的，这意味着Gabor密度定理对于无理性格类型的子空间表示是有效的。参考文献
It is well known that a Gabor representation onadmits a frame generatorif and only if the associated lattice satisfies the Beurling density condition, which in turn can be characterized as the “trace condition” for the associated von Neumann algebra. It happens that this trace condition is also necessary for any projective unitary representation of a countable group to admit a frame vector. However, it is no longer sufficient for general representations, and in particular not sufficient for Gabor representations when they are restricted to proper time-frequency invariant subspaces. In this short note we show that the condition is also sufficient for a large class of projective unitary representations, which implies that the Gabor density theorem is valid for subspace representations in the case of irrational types of lattices. References