论文标题
分析Weyl构造的动态cartan sublgebras
Analyzing the Weyl construction for dynamical Cartan subalgebras
论文作者
论文摘要
当减少扭曲的$ c^*$ - 代数$ c^*_ r（\ Mathcal {g}，c）$的非主要群体类固醇$ \ Mathcal {g} $接纳了雷诺的cartan subalgebra，雷诺在cartan subgebras上的工作暗示了另一个cartan subgebras的工作，暗示了$ c^*c^*c^c^*c carte cartan subgebra。在与Reznikoff和Wright的较早论文中，我们确定了这种cartan subergebra的情况，是由$ \ Mathcal {g} $的亚组$ \ Mathcal {s} $产生的。在本文中，我们研究了原始groupoids $ \ mathcal {s}，\ mathcal {g} $与weyl clastoid和与cartan对相关的扭曲之间的关系。我们首先确定Cartan subalgebra $ c^*_ r（\ Mathcal {s}，c）$的频谱$ \ mathfrak {b} $。然后，我们表明商groupoid $ \ mathcal {g}/\ mathcal {s} $都在$ \ mathfrak {b} $上作用，并且相应的动作groupoid正是cartan对的weyl groupoid。最后，我们证明，如果商映射$ \ MATHCAL {G} \ to \ MATHCAL {G}/\ MATHCAL {s} $接收连续的部分，那么Weyl Twist也通过$ 2 $ 2 $ -COCYCLE $ 2 $ -COCYCLE给出了$ \ Mathcal {g}/\ Mathcal calcal calcal calcal {
When the reduced twisted $C^*$-algebra $C^*_r(\mathcal{G}, c)$ of a non-principal groupoid $\mathcal{G}$ admits a Cartan subalgebra, Renault's work on Cartan subalgebras implies the existence of another groupoid description of $C^*_r(\mathcal{G}, c)$. In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid $\mathcal{S}$ of $\mathcal{G}$. In this paper, we study the relationship between the original groupoids $\mathcal{S}, \mathcal{G}$ and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum $\mathfrak{B}$ of the Cartan subalgebra $C^*_r(\mathcal{S}, c)$. We then show that the quotient groupoid $\mathcal{G}/\mathcal{S}$ acts on $\mathfrak{B}$, and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly we show that, if the quotient map $\mathcal{G}\to\mathcal{G}/\mathcal{S}$ admits a continuous section, then the Weyl twist is also given by an explicit continuous $2$-cocycle on $\mathcal{G}/\mathcal{S} \ltimes \mathfrak{B}$.
