On a dimension for a class of homeomorphism groups
On a dimension for a class of homeomorphism groups
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关于一类同胚群的维数
DOI：
10.1007/bf01420115
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发表时间：
1980
影响因子：
1.4
通讯作者：
Wolfgang Krieger
中科院分区：
文献类型：
--
作者：
Wolfgang Krieger
维数已被证明是冯·诺依曼代数分类论以及测度空间自同构群分类论中非常有用的工具。至于 C* 代数，Elliott 最近提出了一种理论，其中包含通过维度 1, 2] 范围对 AF 代数进行分类。紧密遵循艾略特的思想，我们在这里引入零维紧致体的一类可数局部有限同胚群的维度。我们认为的群体称为充足群体。我们要求这样一个充足的群，它的每个元素都有一个开放的不动点集，并且可以从该群的元素拼凑在一起的每个同胚都已经在该群中。这种充足群的概念让人想起由 Dye I-1 提出的遍历理论中的完整群的概念。充足群的维数范围是有序交换群的段，即它的维数群。调用作用于零维紧致体 X 的可数同构群 f# 和 f~ 以及作用于零维紧致体 X 的 f~，空间同构，如果存在 X 到）~ 的同态 h，使得 f~= hfgh-1。我们将看到，当且仅当它们的维数范围是同构的（第 3 节）时，大量的群才是空间同构的。我们应该注意到，对于第二个可数a-紧零维空间也有模拟理论。
The dimension has proved to be a highly useful tool in the classification theory of von Neumann algebras and also in the classification theory of automorphism groups of measure spaces. As for C*-algebras, Elliott has recently presented a theory that contains a classification of the AF-algebras by means of the ranges of their dimensions 1, 2]. Closely adhering to Elliott's ideas we introduce here a dimension for a class of countable locally finite homeomorphism groups of a zero dimensional compactum. The groups we consider we call ample groups. We require of such an ample group that every one of its elements has an open fixed point set, and that every homeomorphism that can be patched together from elements of the group is already in the group. This notion of an ample group is reminiscent of the notion of a full group in ergodic theory as initiated by Dye I-1]. The range of the dimension of an ample group is the segment of an ordered abelian group, its dimension group. Call countable homeomorphism groups f# and f~, acting on a zero-dimensional compactum X and f~ acting on a zero-dimensional compactum)~, spatially isomorphic if there is a homeomorphism h of X onto)~ such that f~= hfgh-1. We shall see that ample groups are spatially isomorphic if and only if the ranges of their dimensions are isomorphic (Sect. 3). We shall note that one has the analog theory also for of second countable a-compact zerodimensional spaces.