Actions by the classical Banach spaces
Actions by the classical Banach spaces
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DOI：
10.2307/2586545
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发表时间：
2000-03
影响因子：
0.6
通讯作者：
G. Hjorth
中科院分区：
文献类型：
--
作者：
G. Hjorth
对连续群体行为的研究在数学中是无处不在的，也许我们能够在ZFC中证明定理的最一般的行为类型是那些波兰群体作用于波兰空间的行为。对于这个一般的类，我们可以找到像[29]这样的作品，它建立在遍历理论的思想之上，并在测度论和拓扑上下文中研究了局部紧群的作用。另一方面，模型理论中的文本，如[12]，通常会考虑与整数的所有排列的对称群的动作有关的问题。更一般地说，[1]表明无限对称群的闭子群诱导的轨道等价关系可以约化为相应的可数模型类上的同构关系。本文考虑了由可分巴拿赫空间在波兰空间上的连续作用所形成的第三类。这些例子不能归入前两个标题，并且从[10]可知，由这些作用产生的商空间的Borel基数与对称群或任何局部紧致波兰群作用所引起的等价关系是不可比较的。首先要解决的问题之一是这些等价关系的复杂性。这个问题出现在b[1]中。
The study of continuous group actions is ubiquitous in mathematics, and perhaps the most general kinds of actions for which we can hope to prove theorems in just ZFC are those where a Polish group acts on a Polish space. For this general class we can find works such as [29] that build on ideas from ergodic theory and examine actions of locally compact groups in both the measure theoretic and topological contexts. On the other hand a text in model theory, such as [12], will typically consider issues bearing on the actions by the symmetric group of all permutations of the integers. More generally [1] shows that the orbit equivalence relations induced by closed subgroups of the infinite symmetric group can be reduced to the isomorphism relation on corresponding classes of countable models. This paper considers a third category formed by the continuous actions of separable Banach spaces on Polish spaces. These examples cannot be subsumed under the two earlier headings, and it is known from [10] that the Borel cardinalities of the quotient spaces that arise from such actions are incomparable with the equivalence relations induced by the symmetric group or any locally compact Polish group action. One of the first things to be addressed concerns the complexity of these equivalence relations. This question for appears in [1].