THE INDEX OF (WHITE) NOISES AND THEIR PRODUCT SYSTEMS
THE INDEX OF (WHITE) NOISES AND THEIR PRODUCT SYSTEMS
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（白）噪声及其产品系统的指数
DOI：
10.1142/s0219025706002573
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
Michael Skeide
中科院分区：
文献类型：
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作者：
Michael Skeide
几乎每一篇关于Arveson系统（即希尔伯特空间的乘积系统）的论文都是从回忆它们的基本分类开始的，为每个Arveson系统分配一个类型和一个索引。因此，很自然地要问，对于希尔伯特模的乘积系统，也可以提出一个类似的分类。然而，尽管类型的定义是简单的，但索引的定义存在障碍。但是，当限制到我们在这里介绍的空间乘积系统的范畴时，所有的障碍都可以被消除，并且在Arveson系统的情况下，它与空间的通常定义相匹配。这并不是真正的损失，因为非空间Arveson系统的索引定义是相当正式的，并不反映空间Arveson系统的索引所携带的信息。E0-半群产生乘积系统。我们对空间乘积系统的定义，即存在一个中心的单位元，与Powers的空间定义相匹配，因为乘积系统所源自的E0-半群允许一个交织等距半群。我们发现，每一个空间的产品系统包含一个唯一的最大完全空间的子系统（由所有单位产生），同构的产品系统的时间顺序Fock模块。（存在由其单位生成的非空间产品系统。因此，它们不可能是Fock模块。）空间乘积系统的指数我们定义为决定Fock模的（唯一）Hilbert双模。为了表明该指数值得命名指数，我们提供了一个产品的产品系统下，该指数是可加的（直和）。虽然对于Arveson系统来说有张量积，但对于一般的乘积系统来说，张量积作为一个乘积系统没有意义。即使对于Arveson系统，我们的乘积一般也只是张量积的一个子系统。此外，它的构造显然取决于其因子的中心参照单位的选择。乘积系统的空间性意味着它可以从具有不变向量期望的E0-半群导出，即从噪声导出。我们将空间乘积系统的乘积推广到噪声的乘积，并研究了它的性质。最后，我们应用我们的技巧来显示Fowler的结果的模模拟，自由流是完全空间的，我们计算它们的指数。
Almost every paper about Arveson systems (i.e. product systems of Hilbert spaces) starts by recalling their basic classification assigning to every Arveson system a type and an index. So it is natural to ask in how far an analogue classification can also be proposed for product systems of Hilbert modules. However, while the definition of type is plain, there are obstacles for the definition of index. But all obstacles can be removed when restricting to the category which we introduce here as spatial product systems and that matches the usual definition of spatial in the case of Arveson systems. This is not really a loss because the definition of index for nonspatial Arveson systems is rather formal and does not reflect the information the index carries for spatial Arveson systems. E0-semigroups give rise to product systems. Our definition of spatial product system, namely, existence of a unital unit that is central, matches Powers' definition of spatial in the sense that the E0-semigroup from which the product system is derived admits a semigroup of intertwining isometries. We show that every spatial product system contains a unique maximal completely spatial subsystem (generated by all units) that is isomorphic to a product system of time ordered Fock modules. (There exist nonspatial product systems that are generated by their units. Consequently, these cannot be Fock modules.) The index of a spatial product system we define as the (unique) Hilbert bimodule that determines the Fock module. In order to show that the index merits the name index we provide a product of product systems under which the index is additive (direct sum). While for Arveson systems there is the tensor product, for general product systems the tensor product does not make sense as a product system. Even for Arveson systems our product is, in general, only a subsystem of the tensor product. Moreover, its construction depends explicitly on the choice of the central reference units of its factors. Spatiality of a product system means that it may be derived from an E0-semigroup with an invariant vector expectation, i.e. from a noise. We extend our product of spatial product systems to a product of noises and study its properties. Finally, we apply our techniques to show the module analogue of Fowler's result that free flows are comletely spatial, and we compute their indices.