Representation and measure theory of infinite dimensional moues and its applications

这一时期的主要课题涉及拓扑群上的正定函数：第一个是将一般局部紧群上正定函数的G-N-S构造方法推广到无限维群上。我们证明了，如果我们有一个准不变的概率测度，其容许移位形成一个Denge子群，则这种扩展也是可能的。这种测度在局部紧群上取代Haar测度起着关键作用。一个有趣的和重要的例子，无限维群，承认这种措施是集团的同胚光滑流形。第二个问题是：当且仅当正定函数的极端分解与酉表示的不可约分解通过通常的局部紧群上的G-N-S构造重合时？到目前为止，我们已经得到了上述问题的一个充分必要条件。然而，它的描述并不那么清楚，我们正在寻找另一个比这一个更具体的条件。当然，问题并不总是肯定的。我们给出了负有趣的例子，其起源是由于托马：在1964年，托马有一个完整的分类的字符，这是由定义正常的痕迹对因素的H型。然后经过约30年的这项工作，小畑了解体的这些字符。因此，我们有一个问题是否解体对应于不可约分解或没有。我们将在这里写一个部分回答这个问题简要：这取决于托马参数的特点，他的描述;它是负的，如果参数采取一些值。它是否总是负的，以及更具体的等价条件的研究是留给我们的主要问题。

The main subjects of this period concerns with positive-definite functions on topological groups :The first one is an extension of the G-N-S construction method for the positive-definite functions on the usual locally compact groups to that for infinite dimensional groups. We showed that the extension is also possible, if we have a quasi-invariant probability measure whose admissible shifts form a denge subgroup. Such a measure plays a key role in place of the Haar measures on the locally compact groups. An interesting and important example of the infinite dimensional groups which admit such measures is the group of diffeomorphisms on smooth manifolds. The detailed arguments are already published in Math Z.The second one is the following problem :When and only when do extreme decompositions of positive-definite functions coincide with irreducible decompositions of unitary representations through the G-N-S construction on the usual locally compact groups ?Up to the present time, we already have obtained a necessary and sufficient condition for the above problem. However its description is not so clear that we are looking for another more concrete condition than that one.Of course, the problem is not always affirmative. We give the negative interesting examples whose origin is due to Thoma :In 1964, Thoma had a complete classification of the characters which is by definition normal traces on factors of type H. Then after about 30 years of this work, Obata had a disintegration of these characters.Hence, we have a problem whether the disintegration corresponds to irreducible decomposition or not.We will write here a partial answer for this problem briefly : It depends on the Thoma parameter which characterize his description ; It is negative if the parameter takes some values. Whether it is always negative or not is left for us, as well as the study of more specific equivalent conditions for the main problem.