Random Sets and Invariants for (Type II) Continuous Tensor Product Systems of Hilbert Spaces
Random Sets and Invariants for (Type II) Continuous Tensor Product Systems of Hilbert Spaces
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希尔伯特空间（II 型）连续张量积系统的随机集和不变量
DOI：
10.1090/memo/0930
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发表时间：
2003
期刊：
影响因子：
--
通讯作者：
V. Liebscher
中科院分区：
文献类型：
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作者：
V. Liebscher
在一系列文件Tsirelson建造的措施类型的随机集和广义随机过程的一个新的范围内的例子连续张量积系统的希尔伯特空间介绍的Arveson分类$E_0$-半群。本文建立了一个匡威：Hilbert空间的每个连续张量积系都有随机（闭）集在[0，1]或$R +$中的分布测度型。这些测量类型是固定的，并且在不相交的区间上分解。在这种构造的特殊情况下，对应的测度类型是乘积系统的不变量，并且刻画了不变量的值域。
此外，在详细研究这类测度类型的基础上，我们为每一个平稳分解测度类型构造了一个连续的Hilbert空间张量积系统，使得这类测度类型作为上述不变量出现.
上述类型的测度类型与相应的$L^\infty$-空间的表示有关。这导致直接积分表示的元素，一个给定的产品系统，联合收割机结合以及张量积。使用这种结构的建设性的方式，我们可以涉及到任何（III型）产品系统的产品系统类型II_0 $保持同构类。因此，第三类产品系统的分类减少到第二类产品系统的分类。
进一步，我们证明了希尔伯特空间的代数连续张量积系统上的所有相容可测结构都产生同构的乘积系统。因此，Hilbert空间的连续张量积系统的可测结构本质上由它的代数结构决定。
In a series of papers Tsirelson constructed from measure types of random sets and generalised random processes a new range of examples for continuous tensor product systems of Hilbert spaces introduced by Arveson for classifying $E_0$-semigroups. This paper establishes the converse: Each continuous tensor product systems of Hilbert spaces comes with measure types of distributions of random (closed) sets in [0,1] or $R_+$. These measure types are stationary and factorise over disjoint intervals. In a special case of this construction, the corresponding measure type is an invariant of the product system and the range of the invariant is characterized.
Moreover, based on a detailed study of this kind of measure types, we construct for each stationary factorizing measure type a continuous tensor product systems of Hilbert spaces such that this measure type arises as the before mentioned invariant.
The measure types of the above described kind are connected with representations of the corresponding $L^\infty$-spaces. This leads to direct integral representations of the elements of a given product system which combine well under tensor products. Using this structure in a constructive way, we can relate to any (type III) product system a product system of type $II_0$ preserving isomorphy classes. Thus, the classification of type III product systems reduces to that of type II ones.
Further, we show that all consistent measurable structures on an algebraic continuous tensor product systems of Hilbert spaces yield isomorphic product systems. Thus the measurable structure of a continuous tensor product systems of Hilbert spaces is essentially determined by its algebraic one.