Research on Inclusion Relations of Subalgebras

在算子代数领域，我们研究了von Neumann子代数包含关系的全局分析。我们明确了在分析中起主要作用的因素，并对不可约包含关系建立了一些约简理论。为此，首先定义了包含关系的几种类型，即半有限型和有限型。其次，给出了Radon-Nikodym导数和算子值权的分解的一些新概念及其性质。对于一个权值为E的算子，从一个von Neumann代数到子代数，我们定义了E的标准对应（或交换子）E'，它满足链式法则的性质。应用这些结果，我们可以得到以下结果。(1)一个链式法则的性质也适用于一个算子值权的指示导数，这个概念是由我们提出的。(2)对于一般von Neumann代数对，如果存在一个指标为有界算子的条件期望，则存在唯一的最小指标型条件期望。这是Hiai结果的延伸。(3)很容易证明，最小类型条件期望的卷积在一般情况下也是最小类型。(4)对于一个关于Pimsner-Popa熵的可加性问题，我们成功地给出了一个完整的答案。在函数代数中，我们明确了解析函数代数的零集、峰值插值集等之间的关系。

In a field of operator algebras, we studied global analysis of inclusion relations of von Neumann subalgebras. We have made it clear what played principal roles on the analysis and have established some reduction theory to irreducible inclusion relations.In order to do so, first of all, we defined some types of inclusion relations, so called, semifinite type and finite type. Secondly, we developed some new notions and their properties on Radon-Nikodym derivative and disintegration of operator valued weights. For an operator valued weight E from a von Neumann algebra onto a subalgebra, we define the standard correspondent (or commutant) E' of E, which satisfies a chain rule property. Applying these results, we can get the following. (1) A chain rule property is also true to hold for indicial derivative of an operator valued weight, whose notion had been developed by us. (2) For a general pair of von Neumann algebras, if there is a conditional expectation whose index is a bounded operator, there exists uniquely the conditional expectation of minimal index type. This is an extension of Hiai's result. (3) It is easy to show that a convolution of conditional expectations of minimal type is also of minimal type in a general setting. (4) For a problem on additivity of Pimsner-Popa's entropy, we have succeeded to give a complete answer.In a field of function algebra, we have made clear some relation among a zero set, a peak interpolation set, and so on for an analytic function algebra.