Representation and measure theory of infinite dimensional moues and its applications

这一时期的主要内容涉及拓扑群上的正定函数：第一个是将G-N-S关于通常局部紧群上的正定函数的构造方法推广到无限维群上的构造方法。我们证明了，如果我们有一个准不变概率度量，其容许移位形成一个稠子群，那么这种扩张也是可能的。在局部紧群上，这种度量取代了Haar度量起到了关键作用。光滑流形上的微分同胚群是无穷维群的一个有趣而重要的例子。第二个问题是：正定函数的极值分解何时且仅当通过通常的局部紧群上的G-N-S构造与么正表示的不可约分解重合？到目前为止，我们已经得到了上述问题的一个充要条件。然而，它的描述并不是那么清楚，我们正在寻找另一个比那个更具体的条件。当然，这个问题并不总是肯定的。我们给出了起源于Thoma的负的有趣的例子：在1964年，Thoma对这些特征进行了完全的分类，根据定义这是H型因子上的正规迹。然后，经过大约30年的工作，Obata有了这些特征的分解。因此，我们有一个问题，这种分解是否对应于不可约分解。我们将在这里简短地写出这个问题的部分答案：它取决于描述他描述的Thoma参数；如果参数取某些值，则为负。它是否总是负的留给我们，以及对主要问题的更具体的等价条件的研究。

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.