量子相对模函数

摘要：

设$\mathbb{H}\trianglelefteq\mathbb{G}$是局部紧量子群的闭正规子群.我们引入了一个与$L^{\infty}(\mathbb{G})$相关的严格正的类群元素，它粗略地测量了$\mathbb{G}$通过共轭作用对$\mathbb{H}$起测量作用的失败。通过与经典情况类比，该元素的平凡性等价于$\mathbb{G}$和$\mathbb{G}/\mathbb{H}$具有相同模元素的条件。如果$\mathbb{H}\le \mathbb{G}$是中心的，则该条件是自动的，并且通常意味着$\mathbb{H}$的单模性。我们还描述了一个双射之间的严格正群元素$δ$隶属于$C_0(\mathbb{G})$和量子群态射$\mathbb{G}\to (\mathbb{R},+)$，与闭合图像的态射很容易描述的频谱方面的$δ$。这意味着性质-（T）局部紧量子群不允许非明显的严格正类群元。

英文摘要：

Let $\mathbb{H}\trianglelefteq\mathbb{G}$ be a closed normal subgroup of a locally compact quantum group. We introduce a strictly positive group-like element affiliated with $L^{\infty}(\mathbb{G})$ that, roughly, measures the failure of $\mathbb{G}$ to act measure-preservingly on $\mathbb{H}$ by conjugation. The triviality of that element is equivalent to the condition that $\mathbb{G}$ and $\mathbb{G}/\mathbb{H}$ have the same modular element, by analogy with the classical situation. This condition is automatic if $\mathbb{H}\le \mathbb{G}$ is central, and in general implies the unimodularity of $\mathbb{H}$. We also describe a bijection between strictly positive group-like elements $δ$ affiliated with $C_0(\mathbb{G})$ and quantum-group morphisms $\mathbb{G}\to (\mathbb{R},+)$, with the closed image of the morphism easily described in terms of the spectrum of $δ$. This then implies that property-(T) locally compact quantum groups admit no non-obvious strictly positive group-like elements.

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