Non-Commutative Stone duality: Inverse Semigroups, Topological Groupoids and C*-Algebras
Non-Commutative Stone duality: Inverse Semigroups, Topological Groupoids and C*-Algebras
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非交换 Stone 对偶性：逆半群、拓扑群形和 C*-代数
DOI：
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发表时间：
2011
影响因子：
0.8
通讯作者：
M. Lawson
中科院分区：
文献类型：
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作者：
M. Lawson
我们研究了Stone对偶的一个非交换推广，它将一类称为布尔逆半群的逆半群与一类称为Hausdorff布尔群胚的拓扑群胚联系起来。本文的大部分内容都是为了证明布尔逆半群是作为我们称之为前布尔的逆半群的完备化而产生的。逆半群是预布尔的当且仅当每个紧滤子是超滤子，其中紧滤子是由Exel和伦茨的思想结合起来定义的。给出了半群是预布尔半群的一个简单的必要条件，并给出了满足该条件的逆半群的例子，从而证明了多圈逆幺半群和多圈上的某些Rees矩阵半群是预布尔半群，并证明了它们的补元的单位群恰是Reppson-Higman群Gn，r.由适当的有向图产生的逆半群也是预布尔的，并且在我们的非交换Stone对偶下由这些图逆半群产生的拓扑群胚是由Cuntz-Krieger C*-代数产生的群胚。我们的理论的一个初等应用表明，有限基本布尔逆半群是有限对称逆幺半群的有限直积。最后，我们解释如何紧密过滤器相关的总理过滤器设置场景为未来的工作。
We study a non-commutative generalization of Stone duality that connects a class of inverse semigroups, called Boolean inverse ∧-semigroups, with a class of topological groupoids, called Hausdorff Boolean groupoids. Much of the paper is given over to showing that Boolean inverse ∧-semigroups arise as completions of inverse ∧-semigroups we call pre-Boolean. An inverse ∧-semigroup is pre-Boolean if and only if every tight filter is an ultrafilter, where tight filters are defined by combining ideas of both Exel and Lenz. A simple necessary condition for a semigroup to be pre-Boolean is derived and a variety of examples of inverse semigroups are shown to satisfy it. Thus the polycyclic inverse monoids, and certain Rees matrix semigroups over the polycyclics, are pre-Boolean and it is proved that the groups of units of their completions are precisely the Thompson–Higman groups Gn, r. The inverse semigroups arising from suitable directed graphs are also pre-Boolean and the topological groupoids arising from these graph inverse semigroups under our non-commutative Stone duality are the groupoids that arise from the Cuntz–Krieger C*-algebras. An elementary application of our theory shows that the finite, fundamental Boolean inverse ∧-semigroups are just the finite direct products of finite symmetric inverse monoids. Finally, we explain how tight filters are related to prime filters setting the scene for future work.