论文标题
反向半群的统一ROE代数
The uniform Roe algebra of an inverse semigroup
论文作者
论文摘要
考虑到一个离散且可计数的反向半群$ s $，一个人可以与组案例相比，其几何方面可以研究。特别是，我们可以在其Schützenberger图的不相交联合中给出的Path Metric给出天然度量。我们用$λ_s$表示该图，继承了$ s $的许多结构。在本文中，我们比较了c*-algebra $ \ mathcal {r} _s $，由左侧定期代表$ \ ell^2（s）$和$ \ ell^2（s）$和$ \ ell^\ eld^\ infty（s）$生成，与统一的Roe algebra与均值$ c^*u（λ__s）。这产生了$ \ MATHCAL {R} _S = C^*_ U（λ_S）$的苦难，该$概括了$ s $的有限生成。我们已经用有限的实验性（FL）称其为此，因为它可以以$λ_s$的方式标记为限制。 图$λ_s$和上面的FL条件还允许分析$λ_s$的大规模属性，并将它们与统一ROE代数的C* - 销售相关联。特别是，我们表明$ s $的域可测量性（概述了半群的概括性的定义，cf.，[5]）是$λ_s$的准时不变性。此外，我们根据相应的c* - 代理的核性和精确性来表征$λ_s$（或其组件）的属性a。我们还从这个大规模的角度来处理F-Inverse和E-Norial逆向半群的特殊类别。
Given a discrete and countable inverse semigroup $S$ one can study, in analogy to the group case, its geometric aspects. In particular, we can equip $S$ with a natural metric, given by the path metric in the disjoint union of its Schützenberger graphs. This graph, which we denote by $Λ_S$, inherits much of the structure of $S$. In this article we compare the C*-algebra $\mathcal{R}_S$, generated by the left regular representation of $S$ on $\ell^2(S)$ and $\ell^\infty(S)$, with the uniform Roe algebra over the metric space, namely $C^*_u(Λ_S)$. This yields a chacterization of when $\mathcal{R}_S = C^*_u(Λ_S)$, which generalizes finite generation of $S$. We have termed this by finite labeability (FL), since it holds when the $Λ_S$ can be labeled in a finitary manner. The graph $Λ_S$, and the FL condition above, also allow to analyze large scale properties of $Λ_S$ and relate them with C*-properties of the uniform Roe algebra. In particular, we show that domain measurability of $S$ (a notion generalizing Day's definition of amenability of a semigroup, cf., [5]) is a quasi-isometric invariant of $Λ_S$. Moreover, we characterize property A of $Λ_S$ (or of its components) in terms of the nuclearity and exactness of the corresponding C*-algebras. We also treat the special classes of F-inverse and E-unitary inverse semigroups from this large scale point of view.
